On tau-functions for the Toda lattice hierarchy

Abstract

We extend a recent result of Dubrovin et al. in On tau-functions for the KdV hierarchy, arXiv:1812.08488 to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions and use them to give explicit formulae for the generating series of k-point correlation functions of the solution. Applications to computing GUE correlators and Gromov–Witten invariants of the Riemann sphere are under consideration.

1 Introduction

The Toda lattice hierarchy, which contains the Toda lattice equation

$$\begin{aligned} \ddot{\sigma }(n) \; = \; e^{\sigma (n-1)-\sigma (n)} \, - \, e^{\sigma (n)-\sigma (n+1)}, \end{aligned}$$

(1)

is an important integrable hierarchy of nonlinear differential–difference equations [18, 19, 22, 27]. In this paper, following the idea of [13], we derive new formulae for generating series of k-point correlation functions for the Toda lattice hierarchy by using the matrix resolvent approach [10] and by introducing a pair of wave functions.

1.1 Toda lattice hierarchy and tau-function

Let

$$\begin{aligned} {{\mathcal {A}}}\, := \, {{\mathbb {Z}}}\, [ v_0,w_0,v_{\pm 1}, w_{\pm 1}, v_{\pm 2}, w_{\pm 2}, \ldots ] \end{aligned}$$

(2)

be the polynomial ring. Define the shift operator $\Lambda :{{\mathcal {A}}}\rightarrow {{\mathcal {A}}}$ via

$$\begin{aligned} \Lambda (1)\; = \;1, \quad \Lambda (v_i) \; = \; v_{i+1}, \quad \Lambda (w_i) \; = \; w_{i+1}, \quad \Lambda (f g) \; = \; \Lambda (f) \, \Lambda (g)\end{aligned}$$

$\forall \, i\in {{\mathbb {Z}}}$ and $f,g\in {{\mathcal {A}}}$. Denote by $\Lambda ^{-1}$ the inverse of $\Lambda$ satisfying $\Lambda ^{-1} (v_i) = v_{i-1}$, $\Lambda ^{-1} (w_i) = w_{i-1}$, and $\Lambda ^{-1} (f g) = \Lambda ^{-1} (f) \, \Lambda ^{-1} (g)$. For a difference operator P on ${{\mathcal {A}}}$, we mean an operator of the form $P = \sum _{m\in {{\mathbb {Z}}}} P_m \, \Lambda ^m$, where $P_m \in {{\mathcal {A}}}$. Denote $P_+:=\sum _{m\ge 0} P_m \, \Lambda ^m$, $P_-:=\sum _{m< 0} P_m \, \Lambda ^m$, ${\mathrm{Coef}}(P,m):=P_m$. A linear operator $D:{{\mathcal {A}}}\rightarrow {{\mathcal {A}}}$ is called a derivation on ${{\mathcal {A}}}$, if

$$\begin{aligned} D(fg) \; = \; D(f) \, g \; + \; f \, D(g), \quad \forall \,f,g\in {{\mathcal {A}}}. \end{aligned}$$

The derivation D is called admissible if it commutes with $\Lambda$. Clearly, every admissible derivation D is uniquely determined by the values $D(v_0)$ and $D(w_0)$. Let

$$\begin{aligned} L\, := \, \Lambda \; + \; v_0 \; + \; w_0 \, \Lambda ^{-1} \end{aligned}$$

(3)

be a difference operator, and define a sequence of difference operators $A_k$, $k\ge 0$ by

$$\begin{aligned} A_k \, := \, \bigl (L^{k+1}\bigr )_+. \end{aligned}$$

(4)

We associate with $A_k$ a sequence of admissible derivations $D_k:{{\mathcal {A}}}\rightarrow {{\mathcal {A}}}$ defined via

$$\begin{aligned} D_k(v_0) \, := \, {\mathrm{Coef}} \bigl ([A_k,L], 0\bigr ), \quad D_k(w_0) \, := \, {\mathrm{Coef}} \bigl ([A_k,L], -1\bigr ), \qquad k\ge 0. \end{aligned}$$

(5)

The first few $D_k(v_0)$ and $D_k(w_0)$ are $D_0(v_0)=w_1-w_0$, $D_0(w_0)=w_0 \, (v_0-v_{-1})$; $D_1(v_0)=w_1(v_1+v_0)-w_0(v_0+v_{-1})$, $D_1(w_0)=w_0\bigl (w_1-w_{-1}+v_0^2-v_{-1}^2\bigr )$, etc.

Lemma 1

The operators $D_k$, $k\ge 0$ pairwise commute.

This lemma was known. We call $D_k$ the Toda lattice derivations, and (5) the abstract Toda lattice hierarchy.

A tau-structure associated with the derivations $(D_k)_{k\ge 0}$ is a collection of polynomials $\bigl (\Omega _{p,q}, S_p\bigr )_{p,q\ge 0}$ in ${\mathcal {A}}$ satisfying

$$\begin{aligned}&\Omega _{p,q} \; = \; \Omega _{q,p}, \quad D_r \bigl (\Omega _{p,q}\bigr ) \; = \; D_q \bigl (\Omega _{p,r}\bigr ), \end{aligned}$$

(6)

$$\begin{aligned}&(\Lambda -1 ) \, \bigl (\Omega _{p,q}\bigr ) \; = \; D_q \bigl (S_p\bigr ) , \end{aligned}$$

(7)

$$\begin{aligned}&w_0 \, \bigl (1 \,-\, \Lambda ^{-1} \bigr ) \bigl (S_p\bigr ) \; = \; D_p (w_0) \, \end{aligned}$$

(8)

for all $p,q,r\ge 0$. It can be shown (e.g., [10]) that the tau-structure exists and is unique up to replacing $\Omega _{p,q},S_p$ by $\Omega _{p,q}+c_{p,q}$ and $S_p+a_p$ respectively, where $c_{p,q}=c_{q,p}$ and $a_ p$ are arbitrary constants. The tau-structure $\Omega _{p,q},S_p$ is called canonical if

$$\begin{aligned} \Omega _{p,q}\big |_{v_i=0, \, w_i=0, \, i\in {{\mathbb {Z}}}} \; = \; 0, \quad S_{p} \big |_{v_i=0, \, w_i=0, \, i\in {{\mathbb {Z}}}} \; = \; 0. \end{aligned}$$

Let us take $\Omega _{p,q},S_p$ the canonical tau-structure. For $m\ge 3$, define

$$\begin{aligned} \Omega _{p_1,\ldots ,p_m} \, := \, D_{p_1} \cdots D_{p_{m-2}} \, \bigl (\Omega _{p_{m-1} p_m} \bigr ) \in {{\mathcal {A}}}, \qquad p_1,\ldots ,p_m\ge 0. \end{aligned}$$

(9)

By (6), we know that the $\Omega _{p_1,\ldots ,p_m}$, $m\ge 2$, are totally symmetric with respect to permutations of the indices $p_1,\ldots ,p_m$. The first few of these polynomials are

$$\begin{aligned}&S_0 \; = \; v_0, \quad S_1 \; = \; w_1+w_0 +v_0^2, \end{aligned}$$

(10)

$$\begin{aligned}&\Omega _{0,0} \; = \; w_0, \quad \Omega _{0,1}\; = \;\Omega _{1,0}\; = \;w_1(v_1+v_0). \end{aligned}$$

(11)

If we think of $v_0$, $w_0$ as two functions v(n), w(n) of n, respectively, and $v_i$, $w_i$ as $v(n+i)$, $w(n+i)$, then the Toda lattice derivations $D_k$ lead to a hierarchy of evolutionary differential–difference equations, called the Toda lattice hierarchy, given by

$$\begin{aligned}&\frac{\partial v(n)}{\partial t_k} \; = \; D_k(v_0) (n), \qquad \frac{\partial w(n)}{\partial t_k} \; = \; D_k(w_0) (n), \end{aligned}$$

(12)

where $k\ge 0$, and the $D_k(v_0) (n), D_k(w_0) (n)$ are defined as $D_k(v_0), D_k(w_0)$ with $v_i$, $w_i$ replaced by $v(n+i)$, $w(n+i)$, respectively. Lemma 1 implies that the flows (12) all commute. So we can solve the whole Toda lattice hierarchy (12) together, which yields solutions of the form $(v=v(n,\mathbf{t}),w=w(n,\mathbf{t}))$. Here, $\mathbf{t}:=(t_0,t_1,\ldots )$ denotes the infinite time vector. Note that the $k=0$ equations read

$$\begin{aligned}&\dot{v}(n) \; = \; w(n+1) \,- \, w(n), \qquad \dot{w}(n) \; = \; w(n) \, \bigl (v(n)-v(n-1)\bigr ), \end{aligned}$$

(13)

which are equivalent to Eq. (1) via the transformation

$$\begin{aligned} w(n) \; = \; e^{\sigma (n-1)-\sigma (n)}, \qquad v(n) \; = \; - {{\dot{\sigma }}}(n). \end{aligned}$$

Here, dot, “ $\dot{}$ ”, is identified with $\partial /\partial t_0$.

Let V be a ring of functions of n closed under shifting n by $\pm 1$. For two given $f(n),g(n)\in V$, consider the initial value problem for (12) with the initial condition:

$$\begin{aligned} v(n,\mathbf{0}) \; = \;f(n), \quad w(n,\mathbf{0})\; = \;g(n). \end{aligned}$$

(14)

The solution $(v(n,\mathbf{t}), w(n,\mathbf{t}))\in V[[\mathbf{t}]]^2$ exists and is unique, which gives the following 1–1 correspondence:

$$\begin{aligned} \bigl \{\hbox {solution } (v,w)\; \hbox {of } (12) \hbox { in } V[[\mathbf{t}]]^2\bigr \} \; \longleftrightarrow \; \bigl \{\hbox {initial data } (f,g) \bigr \}. \end{aligned}$$

(15)

Example 1

$f(n)=0,~g(n)=n$. (For this case, one can take $V={{\mathbb {Q}}}[n]$.) The corresponding unique solution governs the enumerations of ribbon graphs in all genera.

Example 2

$f(n)= (n+\frac{1}{2})\epsilon , ~ g(n)=1$. (For this case, one can take $V={{\mathbb {Q}}}[\epsilon ][n]$.) The corresponding unique solution governs the Gromov–Witten invariants of ${\mathbb {P}}^1$ in the stationary sector in all genera and all degrees.

Let $(v,w)\in V[[\mathbf{t}]]^2$ be an arbitrary solution to the Toda lattice hierarchy (12). Write $\Omega _{p,q}(n,\mathbf{t})$ and $S_p(n,\mathbf{t})$ as the images of $\Omega _{p,q}$ and $S_p$ under the substitutions

$$\begin{aligned} v_i\mapsto v(n+i,\mathbf{t}),\quad w_i\mapsto w(n+i,\mathbf{t}),\qquad i\in {{\mathbb {Z}}}, \end{aligned}$$

(16)

respectively. (Similar notations will be used for other elements of ${{\mathcal {A}}}$.) Equalities (6) then imply the existence of a function $\tau =\tau (n,\mathbf{t})$ such that for $p,q\ge 0$,

$$\begin{aligned}&\Omega _{p,q}(n,\mathbf{t}) \; = \; \frac{\partial ^2 \log \tau (n,\mathbf{t})}{\partial t_p \partial t_q }, \end{aligned}$$

(17)

$$\begin{aligned}&S_p(n,\mathbf{t}) \; = \; \frac{\partial }{\partial t_p}\log \frac{\tau (n+1,\mathbf{t})}{\tau (n,\mathbf{t})}, \end{aligned}$$

(18)

$$\begin{aligned}&w(n,\mathbf{t}) \; = \; \frac{\tau (n+1,\mathbf{t}) \, \tau (n-1,\mathbf{t})}{\tau (n,\mathbf{t})^2} . \end{aligned}$$

(19)

We call $\tau (n,\mathbf{t})$ the Dubrovin–Zhang (DZ)-type tau-function [10, 15] of the solution (v, w), in short the tau-function of the solution. The symmetry in (9) is more obvious: the image $\Omega _{p_1,\ldots ,p_m}(n,\mathbf{t})$ of $\Omega _{p_1,\ldots ,p_m}$ under (16) satisfies

$$\begin{aligned} \Omega _{p_1,\ldots ,p_m}(n,\mathbf{t}) \; = \; \frac{\partial ^m \log \tau (n,\mathbf{t})}{\partial t_{p_1} \ldots \partial t_{p_m}}, \; \qquad m\ge 2,\; p_1,\ldots ,p_m\ge 0. \end{aligned}$$

(20)

Define $\Omega _p(n,\mathbf{t})=\partial _{t_p} \log \tau (n,\mathbf{t})$, $p\ge 0$. These logarithmic derivatives of $\tau (n,\mathbf{t})$ are called correlation functions of the solution (v, w). The specializations $\Omega _{p_1,\ldots ,p_m}(n,\mathbf{0})$ are called m-point partial correlation functions of (v, w).

Remark 1

The tau-function $\tau (n,\mathbf{t})$ of the solution (v, w) is unique up to multiplying it by the exponential of a linear function of $n,t_0,t_1,t_2,\ldots$.

1.2 Matrix resolvent

The matrix resolvent (MR) method for computing correlation functions for integrable hierarchies was introduced in [1,2,3], and was extended to the discrete case in [10] (in particular to the Toda lattice hierarchy). Denote

$$\begin{aligned} U(\lambda )\, := \, \begin{pmatrix} v_0-\lambda &{}\quad w_0\\ -1 &{}\quad 0\end{pmatrix}. \end{aligned}$$

The following lemma for the Toda lattice hierarchy was proven in [10].

Lemma 2

[10] There exists a unique series $R(\lambda )\in {\mathrm{Mat}}\left( 2,{{\mathcal {A}}}\left[ \left[ \lambda ^{-1}\right] \right] \right)$ satisfying

$$\begin{aligned}&\Lambda \bigl (R(\lambda )\bigr ) \, U(\lambda ) \,-\,U(\lambda ) \, R(\lambda ) \; = \; 0, \end{aligned}$$

(21)

$$\begin{aligned}&{\mathrm{Tr}} \, R (\lambda ) \; = \; 1, \quad \det R(\lambda ) \; = \; 0, \end{aligned}$$

(22)

$$\begin{aligned}&R(\lambda ) \,-\, \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{} \quad 0 \end{pmatrix} \; \in \; {\mathrm{Mat}}\left( 2,{{\mathcal {A}}}\left[ \left[ \lambda ^{-1}\right] \right] \lambda ^{-1}\right) . \end{aligned}$$

(23)

The unique series $R(\lambda )$ in Lemma 2 is called the basic matrix resolvent. The first few terms of $R(\lambda )$ are given by

$$\begin{aligned}&R(\lambda )\; = \; \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 0\end{pmatrix} \; + \; \begin{pmatrix} 0 &{}\quad -w_0 \\ 1 &{}\quad 0\end{pmatrix} \frac{1}{\lambda } \; + \; \begin{pmatrix} w_0 &{}\quad -v_0w_0 \\ v_{-1} &{}\quad -w_0\end{pmatrix} \frac{1}{\lambda ^2} \nonumber \\&\qquad \qquad \qquad \; + \; \begin{pmatrix} w_0(v_0+v_{-1}) &{}\quad -w_0(w_0+w_1+v_0^2) \\ w_0+w_{-1}+v_{-1}^2 &{}\quad -w_0(v_0+v_{-1})\end{pmatrix} \frac{1}{\lambda ^3} \; + \;\cdots . \end{aligned}$$

(24)

Proposition 1

[10] For any $k\ge 2$, the following formula holds true:

$$\begin{aligned} \sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}} \; = \; - \sum _{\pi \in {\mathcal {S}}_k/C_k} \frac{{\mathrm{tr}}\, \prod _{j=1}^k R\bigl (\lambda _{\pi (j)}\bigr )}{\prod _{j=1}^k \bigl (\lambda _{\pi {(j)}}-\lambda _{\pi {(j+1)}}\bigr )} \,-\, \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2}, \end{aligned}$$

(25)

where ${\mathcal {S}}_k$ denotes the symmetry group and $C_k$ the cyclic group, and $\pi (k+1):=\pi (1)$.

The meaning of (25) is the following: For any fixed permutation $(j_1,\ldots ,j_k)$ of $(1,\ldots ,k)$, expanding the right-hand side with respect to $|\lambda _{j_1}|>\cdots>|\lambda _{j_k}|>>0$ gives identical formal power series with the left-hand side. This is because, after the summation over the ${\mathcal {S}}_k/C_k$ and subtracting $\frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2}$, the poles in the diagonal cancel (cf. Proposition 2 of [12] for a straightforward proof of this point). We note that, as formal power series, the coefficients of the both sides of (25) are in ${{\mathcal {A}}}$. We give in Sect. 2 a new proof of (25), where we keep all derivations with coefficients in ${{\mathcal {A}}}$.

1.3 From wave functions to correlation functions

In [13], we introduced the notion of a tuple of wave functions (in many cases a pair) to the study of tau-function without using the Sato theory. Let us generalize it to the Toda lattice hierarchy. Our definition of a pair will be based on the standard construction of wave functions for the Toda lattice hierarchy [5, 6, 27]. For given (f(n), g(n)) a pair of arbitrary elements in V, let L be the linear difference operator $L=\Lambda +f(n)+g(n) \Lambda ^{-1}$. Denote

$$\begin{aligned} s(n) \, := \, - \bigl (1-\Lambda ^{-1}\bigr )^{-1} \bigl (\log g(n)\bigr ). \end{aligned}$$

(26)

The function s(n) is in a certain extension  ${\widehat{V}}$ of V and is uniquely determined by $\log g(n)$ up to a constant. Below we fix a choice of s(n). An element $\psi _A(\lambda ,n)=\bigl (1+{\mathrm{O}}\left( \lambda ^{-1}\right) \bigr ) \,\lambda ^n$ in the module ${{\widetilde{V}}}\left[ \left[ \lambda ^{-1} \right] \right] \lambda ^n$ is called a (formal) wave function of type A associated with f(n), g(n), if $L \bigl ( \psi _A(\lambda ,n)\bigr )= \lambda \psi _A(\lambda ,n)$. Here, $\widetilde{V}$ is a ring of functions of n satisfying

$$\begin{aligned} V\subset (\Lambda -1)\bigl ({\widetilde{V}}\bigr ) \subset {\widetilde{V}}. \end{aligned}$$

An element $\psi _B(\lambda ,n)=\bigl (1+{\mathrm{O}}\bigl (\lambda ^{-1}\bigr )\bigr ) \, e^{-s(n)} \lambda ^{-n}$ in the module ${\widetilde{V}} \left[ \left[ \lambda ^{-1} \right] \right] e^{-s(n)} \lambda ^{-n}$ is called a (formal) wave function of type B, if $L\bigl ( \psi _B(\lambda ,n) \bigr ) = \lambda \psi _B(\lambda ,n)$. Let $\psi _A \in {\widetilde{V}}\left[ \left[ \lambda ^{-1}\right] \right] \, \lambda ^n$ and $\psi _B \in {\widetilde{V}} \left[ \left[ \lambda ^{-1} \right] \right] e^{-s(n)} \lambda ^{-n}$ be two wave functions of type A and of type B associated with (f(n), g(n)), respectively. Define

$$\begin{aligned} d(\lambda ,n) \, := \, \psi _A(\lambda ,n) \, \psi _B(\lambda ,n-1) - \psi _B(\lambda ,n) \, \psi _A(\lambda ,n-1). \end{aligned}$$

(27)

We call $\psi _A,\psi _B$ form a pair if the following normalization condition holds:

$$\begin{aligned} e^{s(n-1)} d(\lambda ,n) \; = \; \lambda . \end{aligned}$$

(28)

The existence of a pair of wave functions is proved in Sect. 3.

Denote by $(v(n,\mathbf{t}),w(n,\mathbf{t}))$ the unique solution in $V[[\mathbf{t}]]^2$ to the Toda lattice hierarchy with (f(n), g(n)) as its initial value, by $\psi _A(\lambda ,n)$ and $\psi _B(\lambda ,n)$ a pair of wave functions associated with (f(n), g(n)) and by $\tau (n,\mathbf{t})$ the DZ-type tau-function of $(v(n,\mathbf{t}),w(n,\mathbf{t}))$. Introduce

$$\begin{aligned} D(\lambda ,\mu ,n) \, := \, \frac{\psi _A(\lambda ,n) \, \psi _B(\mu ,n-1) \,-\, \psi _A(\lambda ,n-1) \,\psi _B(\mu ,n)}{\lambda -\mu }. \end{aligned}$$

(29)

Theorem 1

Fix $k\ge 2$ being an integer. The generating series of k-point partial correlation functions has the following expression:

$$\begin{aligned}&\sum _{i_1,\ldots ,i_k\ge 0} \frac{\partial ^k \log \tau }{\partial t_{i_1} \ldots \partial t_{i_k}} (n,\mathbf{0}) \, \frac{1}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}} \nonumber \\&\qquad \qquad \; = \; (-1)^{k-1} \frac{e^{k s(n-1)}}{\prod _{j=1}^k \lambda _{j}} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k D(\lambda _{\pi (j)},\lambda _{\pi (j+1)},n) \,-\, \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2} . \end{aligned}$$

(30)

Theorem 1 gives an algorithm with the initial value (f(n), g(n)) as the only input for computing the $k_{{\mathrm{th}}}$-order logarithmic derivatives of the tau-function $\tau (n,\mathbf{t})$ evaluated at $\mathbf{t}=\mathbf{0}$ for $k\ge 2$. Indeed, by solving the spectral problem $L(\psi )=\lambda \psi$ with $L=\Lambda +f(n)+g(n)\Lambda ^{-1}$ and with the normalization condition (28), one constructs a pair of wave functions; the coefficients in the ${\mathbf{t}}$-expansion of $\log \tau (n,\mathbf{t})$ are then obtained through algebraic manipulations by using (85). (Recall that in the inverse scattering method (cf., e.g., [18, 19]), an additional integral equation needs to be solved.) Two applications of Theorem 1 are given in Sect. 5. For a certain class of bispectral solutions (cf. [20]), it would be possible to give a canonical way of constructing a pair of wave functions, which was briefly mentioned in [13] for the KdV hierarchy; we plan to do this for KdV and for Toda lattice in a future publication.

Organization of the paper In Sect. 2, we review the MR method of studying tau-structure for the Toda lattice hierarchy. In Sect. 3, we prove the existence of a pair of wave functions. In Sect. 4, we prove Theorem 1 and several other theorems. Applications to the computations of GUE correlators and Gromov–Witten invariants of ${\mathbb {P}}^1$ are given in Sect. 5. In Appendix A, we give an extension of ${{\mathcal {A}}}$, define a pair of abstract pre-wave functions, and prove an abstract version for Theorem 1.

2 Matrix resolvent and tau-structure

We continue in this section with more details in reviewing the MR method [10] to the Toda lattice hierarchy. Denote by ${\mathcal {L}}$ the matrix Lax operator for the Toda lattice:

$$\begin{aligned} {\mathcal {L}}\, := \, \begin{pmatrix} \Lambda &{}\quad 0\\ 0 &{}\quad \Lambda \end{pmatrix} \; + \; \begin{pmatrix} v_0-\lambda &{}\quad w_0\\ -1 &{}\quad 0\end{pmatrix} \; = \; \Lambda \; + \; U(\lambda ). \end{aligned}$$

Let $R(\lambda )$ be the basic matrix resolvent (of ${\mathcal {L}}$). Write

$$\begin{aligned}&R(\lambda ) \; = \; \begin{pmatrix} 1+\alpha (\lambda ) &{}\quad \beta (\lambda ) \\ \gamma (\lambda ) &{}\quad -\alpha (\lambda ) \end{pmatrix}, \end{aligned}$$

(31)

$$\begin{aligned}&\alpha (\lambda ) \; = \; \sum _{i\ge 0} \frac{a_i}{\lambda ^{i+1}}, \quad \beta (\lambda ) \; = \; \sum _{i\ge 0} \frac{b_i}{\lambda ^{i+1}}, \quad \gamma (\lambda ) \; = \; \sum _{i\ge 0} \frac{c_i}{\lambda ^{i+1}}, \end{aligned}$$

(32)

where $a_i,b_i,c_i\in {{\mathcal {A}}}$. From the defining Eqs. (21)–(23), we see that the series $\alpha ,\beta ,\gamma$ satisfy the equations

$$\begin{aligned}&\beta (\lambda ) \; = \; -w_0 \, \Lambda \bigl (\gamma (\lambda )\bigr ), \end{aligned}$$

(33)

$$\begin{aligned}&\gamma (\lambda ) \; = \; \frac{1\; + \;\alpha (\lambda ) \; + \; \Lambda ^{-1} \bigl (\alpha (\lambda )\bigr )}{\lambda - v_{-1}}, \end{aligned}$$

(34)

$$\begin{aligned}&\Bigl (\alpha (\lambda )- \Lambda \bigl (\alpha (\lambda )\bigr )\Bigr )(\lambda -v_0) \,-\, w_0 \, \frac{1+\alpha (\lambda )+\Lambda ^{-1}\bigl (\alpha (\lambda )\bigr )}{\lambda -v_{-1}} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \quad \qquad \quad \quad + w_1 \, \frac{1+\Lambda \bigl (\alpha (\lambda )\bigr )+\Lambda ^2 \bigl (\alpha (\lambda )\bigr )}{\lambda -v_1} \; = \;0, \end{aligned}$$

(35)

$$\begin{aligned}&\alpha (\lambda )+\alpha (\lambda )^2 \; + \; \beta (\lambda ) \gamma (\lambda ) \; = \; 0. \end{aligned}$$

(36)

These equalities give rise to the following recursion relation for $a_i,b_i,c_i$:

$$\begin{aligned}&b_j\; = \; -w_0 \, \Lambda (c_{j}), \qquad c_{j+1} \; = \; v_{-1} \, c_{j} \; + \; \bigl (1 \; + \; \Lambda ^{-1}\bigr ) \, (a_j),\end{aligned}$$

(37)

$$\begin{aligned}&\bigl (1-\Lambda \bigr ) \, (a_{j+1}) \; + \; v_0 \, \bigl (\Lambda -1\bigr ) \, (a_j) \; + \; w_1 \, \Lambda ^2 (c_{j}) \,-\, w_0 \, c_{j} \; = \; 0, \end{aligned}$$

(38)

$$\begin{aligned}&a_{\ell }= \sum _{i+j=\ell -1} \Bigl (w_0 \, c_{i} \, \Lambda (c_j) \,-\, a_{i} \, a_{j} \Bigr ) \end{aligned}$$

(39)

along with

$$\begin{aligned} a_0\; = \;0, \qquad c_0\; = \;1. \end{aligned}$$

(40)

Equations (37)–(40) are called the matrix resolvent recursion relation.

It was proven [10] that the abstract Toda lattice hierarchy (5) can be equivalently written as

$$\begin{aligned}&D_j \, (v_0) \; = \; \bigl (\Lambda -1\bigr ) \, (a_{j+1}), \\&D_j \, (w_0) \; = \; w_0 \, \bigl (\Lambda -1\bigr )\,(c_{j+1}), \end{aligned}$$

where $j\ge 0$. Define an operator $\nabla (\lambda )$ by

$$\begin{aligned} \nabla (\lambda ) \, := \, \sum _{j\ge 0} \frac{D_j}{\lambda ^{j+2}}. \end{aligned}$$

(41)

We have

$$\begin{aligned}&\nabla (\lambda ) \, ( v_0 ) \; = \; \bigl (\Lambda -1\bigr ) \, \bigl (\alpha (\lambda )\bigr ), \end{aligned}$$

(42)

$$\begin{aligned}&\nabla (\lambda ) \, ( w_0 ) \; = \; w_0 \, \bigl (\Lambda -1\bigr ) \, \bigl (\gamma (\lambda )-1\bigr ). \end{aligned}$$

(43)

Lemma 3

There exists a unique element $W(\lambda ,\mu )$ in ${{\mathcal {A}}}\otimes {\mathrm{sl}}_2({{\mathbb {C}}}) \left[ \left[ \lambda ^{-1},\mu ^{-1}\right] \right] \lambda ^{-1} \mu ^{-1}$ of the form

$$\begin{aligned} W(\lambda ,\mu ) \; = \; \begin{pmatrix} X(\lambda ,\mu ) &{}\quad Y(\lambda ,\mu ) \\ Z(\lambda ,\mu ) &{}\quad -X(\lambda ,\mu )\end{pmatrix} \end{aligned}$$

satisfying the following linear inhomogeneous equations for the entries of W:

$$\begin{aligned}&\Lambda \bigl (W(\lambda ,\mu )\bigr ) \, U(\lambda ) \,-\, U(\lambda ) \, W(\lambda ,\mu ) \; + \; \Lambda \bigl (R(\lambda )\bigr ) \, \nabla (\mu ) \bigl (U(\lambda )\bigr ) \nonumber \\&\quad \,-\, \nabla (\mu ) \bigl (U(\lambda )\bigr ) \, R(\lambda ) \; = \;0, \end{aligned}$$

(44)

$$\begin{aligned}&X(\lambda ,\mu ) \; + \; 2 \alpha (\lambda ) \, X(\lambda ,\mu ) \; + \; \gamma (\lambda ) \, Y(\lambda ,\mu ) \; + \; \beta (\lambda ) \, Z(\lambda ,\mu ) \; = \; 0. \end{aligned}$$

(45)

Proof

The existence part of this lemma follows from Lemma 2. Indeed, if we define

$$\begin{aligned} W(\lambda ,\mu ) \, := \, \nabla (\mu ) \bigl (R(\lambda )\bigr ), \end{aligned}$$

then $W(\lambda ,\mu )$ satisfies (44)–(45). To see the uniqueness part, we first note that the (1,2)-entry and the (2,1)-entry of the matrix equation (44) imply that Y and Z can be uniquely expressed in terms of X. Indeed, we have

$$\begin{aligned}&Z(\lambda ,\mu ) \; = \; \tfrac{(1 + \Lambda ^{-1}) (X(\lambda ,\mu ))}{\lambda - v_{-1}} \; + \; \gamma (\lambda ) \tfrac{\Lambda ^{-1} \circ \nabla (\mu ) (v_0)}{\lambda - v_{-1}}, \end{aligned}$$

(46)

$$\begin{aligned}&Y(\lambda ,\mu ) \; = \; - \nabla (\mu ) (w_0) \, \tfrac{1\; + \;\alpha (\lambda ) \; + \; \Lambda (\alpha (\lambda )) }{\lambda - v_{0}} \,- \, w_0 \, \tfrac{(1+ \Lambda ) (X(\lambda ,\mu ))}{\lambda - v_{0}} \; + \; \beta (\lambda ) \tfrac{\nabla (\mu ) (v_0)}{\lambda - v_{0}} . \end{aligned}$$

(47)

Substituting these two expressions in (45), we obtain the following linear inhomogeneous difference equation for X:

$$\begin{aligned}&\Bigl (1 + 2 \alpha (\lambda ) + \tfrac{\beta (\lambda )}{\lambda - v_{-1}} - \tfrac{w_0 \gamma (\lambda )}{\lambda - v_{0}} \Bigr ) X(\lambda ,\mu ) \,- \, \tfrac{ w_0 \gamma (\lambda ) }{\lambda - v_{0}} \Lambda \bigl (X(\lambda ,\mu )\bigr ) \nonumber \\&\quad + \tfrac{\beta (\lambda )}{\lambda - v_{-1}} \Lambda ^{-1} \bigl (X(\lambda ,\mu )\bigr ) \nonumber \\&\quad \; = \; \Bigl (1+\alpha (\lambda ) + \Lambda \bigl (\alpha (\lambda )\bigr )\Bigr ) \gamma (\lambda ) \tfrac{ \nabla (\mu ) (w_0) }{\lambda - v_{0}} \,-\, \beta (\lambda ) \gamma (\lambda ) \bigl (1+\Lambda ^{-1}\bigr ) \Bigl (\tfrac{\nabla (\mu ) (v_0)}{\lambda - v_{0}} \Bigr ). \end{aligned}$$

(48)

Suppose this equation has two solutions $X_1,X_2$ in ${{\mathcal {A}}}\left[ \left[ \lambda ^{-1},\mu ^{-1}\right] \right] \lambda ^{-1} \mu ^{-1}$. Let $X_0=X_1-X_2$, then $X_0\in {{\mathcal {A}}}\left[ \left[ \lambda ^{-1},\mu ^{-1}\right] \right] \lambda ^{-1} \mu ^{-1}$, and it satisfies the following equation:

$$\begin{aligned}&\Bigl (1 + 2 \alpha (\lambda ) + \tfrac{\beta (\lambda )}{\lambda - v_{-1}} - \tfrac{w_0 \gamma (\lambda )}{\lambda - v_{0}} \Bigr ) X_0(\lambda ,\mu ) \,- \, \tfrac{ w_0 \gamma (\lambda ) }{\lambda - v_{0}} \Lambda \bigl (X_0(\lambda ,\mu )\bigr ) \nonumber \\&\quad \; + \; \tfrac{\beta (\lambda )}{\lambda - v_{-1}} \, \Lambda ^{-1} \bigl (X_0(\lambda ,\mu )\bigr ) \; = \; 0. \end{aligned}$$

(49)

It follows that $X_0$ vanishes. Indeed, write $X_0=\sum _{j\ge 0} X_{0,j}(\mu ) \lambda ^{-(j+1)}$. Observe that

$$\begin{aligned} \tfrac{1}{\lambda -v_{m}} \; = \; \tfrac{1}{\lambda }\; + \; \tfrac{v_m}{\lambda ^2} \; + \; \cdots \; \in \; {{\mathcal {A}}}\left[ \left[ \lambda ^{-1}\right] \right] \lambda ^{-1}, \qquad m=-1,0, \end{aligned}$$

and recall that $\alpha (\lambda ),\beta (\lambda ),\gamma (\lambda )\in {{\mathcal {A}}}\left[ \left[ \lambda ^{-1}\right] \right] \lambda ^{-1}$. Then, by comparing the coefficients of powers of $\lambda ^{-1}$ consecutively, we find that $X_{0,0}(\mu )=0$, $X_{0,1}(\mu )=0$, $X_{0,2}(\mu )=0$, $\ldots$. So $X_0=0$. Hence, $X_1=X_2$. The lemma is proved. $\square$

Based on this lemma, we now give a new proof for the following proposition.

Proposition 2

[10] The following equation holds true:

$$\begin{aligned} \nabla (\mu ) \, R(\lambda ) \; = \; \frac{1}{\mu -\lambda } \bigl [ R(\mu ), R(\lambda ) \bigr ] + \bigl [Q(\mu ),R(\lambda )\bigr ], \end{aligned}$$

(50)

where

$$\begin{aligned} Q(\mu ) \, := \, -\frac{{\mathrm{id}}}{\mu } \; + \; \begin{pmatrix} 0 &{} \quad 0 \\ 0 &{} \quad \gamma (\mu ) \\ \end{pmatrix}. \end{aligned}$$

Proof

Define $W^*$ as the right-hand side of (50), i.e.,

$$\begin{aligned} W^*\, := \, \frac{1}{\mu -\lambda } \bigl [ R(\mu ), R(\lambda ) \bigr ] + \bigl [Q(\mu ),R(\lambda )\bigr ]. \end{aligned}$$

More precisely, the entries of $W^*$ have the expressions:

$$\begin{aligned} X^*&= \tfrac{w_0}{\mu -\lambda } \Bigl (\tfrac{(\alpha (\lambda )+\Lambda (\alpha (\lambda ))+1) (\Lambda ^{-1}(\alpha (\mu ))+\alpha (\mu )+1)}{(\lambda -v_0) (\mu -v_{-1})} - \tfrac{(\Lambda ^{-1}(\alpha (\lambda ))+\alpha (\lambda )+1) (\alpha (\mu )+\Lambda (\alpha (\mu ))+1)}{(\lambda -v_{-1}) (\mu -v_0 )}\Bigr ),\end{aligned}$$

(51)

$$\begin{aligned} Y^*&= \tfrac{w_0}{\lambda -\mu } \Bigl (\tfrac{(\alpha (\lambda )+\Lambda (\alpha (\lambda ))+1) (\Lambda ^{-1}(\alpha (\mu )) (\lambda -\mu )+\alpha (\mu ) (\lambda +\mu -2 v_{-1})+\lambda -v_{-1})}{(\lambda -v_0) (\mu -v_{-1})}\nonumber \\&\quad +\tfrac{(2 \alpha (\lambda )+1) (\alpha (\mu )+\Lambda (\alpha (\mu ))+1)}{v_0-\mu }\Bigr ),\end{aligned}$$

(52)

$$\begin{aligned} Z^*&= \tfrac{1}{\lambda -\mu } \Bigl ( \tfrac{(\Lambda ^{-1}(\alpha (\lambda ))+\alpha (\lambda )+1) (\Lambda ^{-1}(\alpha (\mu ))-\alpha (\mu ))}{v_{-1}-\lambda } +\tfrac{(\Lambda ^{-1}(\alpha (\lambda ))-\alpha (\lambda )) (\Lambda ^{-1}(\alpha (\mu ))+\alpha (\mu )+1)}{\mu -v_{-1}} \Bigr ). \end{aligned}$$

(53)

We can then verify that $W^* \in {{\mathcal {A}}}\otimes {\mathrm{sl}}_2({{\mathbb {C}}}) \left[ \left[ \lambda ^{-1},\mu ^{-1}\right] \right] \lambda ^{-1} \mu ^{-1}$, as well as that $W:=W^*$ satisfies Eqs. (44), (45). The latter is done by a lengthy but straightforward calculation. The proposition is proved due to Lemma 3. $\square$

If we define ${{\widetilde{\Omega }}}_{i,j}, {\widetilde{S}}_i$ by

$$\begin{aligned}&\sum _{i,j\ge 0} \frac{{{\widetilde{\Omega }}}_{i,j}}{\lambda ^{i+2} \mu ^{j+2}} \; = \; \frac{{\mathrm{Tr}}\, \bigl (R(\lambda ) R(\mu ) \bigr )}{(\lambda -\mu )^2} \,-\, \frac{1}{(\lambda _1-\lambda _2)^2},\end{aligned}$$

(54)

$$\begin{aligned}&\Lambda \bigl (\gamma (\lambda )\bigr ) \; = \; \lambda ^{-1}\; + \;\sum _{i\ge 0} {\widetilde{S}}_i \, \lambda ^{-i-2}, \end{aligned}$$

(55)

then according to [10], ${{\widetilde{\Omega }}}_{i,j}, {\widetilde{S}}_i$ gives the canonical tau-structure for the Toda lattice, i.e.,

$$\begin{aligned} {{\widetilde{\Omega }}}_{i,j}\; = \;\Omega _{i,j}, \quad {\widetilde{S}}_i\; = \;S_i. \end{aligned}$$

These equalities together with Proposition 2 lead to Proposition 1; see [10] for the detailed proof of Proposition 1.

Before ending this section, we will make two remarks. The first remark is that all the entries of $R(\lambda )$ can be expressed by the canonical tau-structure. Indeed, we have

$$\begin{aligned}&\alpha (\lambda ) \; = \; \sum _{p\ge 0} \Omega _{p,0} \, \lambda ^{-p-2}, \quad \beta (\lambda )=-w_0 \, \Lambda \bigl (\gamma (\lambda )\bigr ), \end{aligned}$$

(56)

$$\begin{aligned}&\Lambda \bigl (\gamma (\lambda )\bigr ) \; = \; \lambda ^{-1}\; + \;\sum _{p\ge 0} S_p \, \lambda ^{-p-2}. \end{aligned}$$

(57)

The proof was in [10]. The second remark is that existence of a tau-structure in general implies Lemma 1, and note that the proof in [10] of the fact that ${{\widetilde{\Omega }}}_{i,j}, {\widetilde{S}}_i$ is a tau-structure does not use the commutativity of the abstract Toda lattice hierarchy, so as a by-product of the matrix resolvent method we get a new proof of Lemma 1 together with a simple construction of the Toda lattice hierarchy. Similar idea was in [3].

3 Pair of wave functions

As in the Introduction, we start with the linear operator $L(n)= \Lambda +f(n) + g(n) \, \Lambda ^{-1}$, where f(n) and g(n) are two given arbitrary elements in V. We show in this section the existence of pairs of wave functions associated with (f(n), g(n)). Let us write

$$\begin{aligned}&\psi _A(\lambda ,n) \; = \; e^{(\Lambda -1)^{-1}y(\lambda ,n)} \lambda ^n , \quad y(\lambda ,n)\, := \, \sum _{i\ge 1} \frac{y_i(n)}{\lambda ^i}, \end{aligned}$$

(58)

$$\begin{aligned}&\psi _B(\lambda ,n) \; = \; e^{(\Lambda -1)^{-1}z(\lambda ,n)} e^{-s(n)} \, \lambda ^{-n}, \quad z(\lambda ,n)\, := \, \sum _{i\ge 1} \frac{z_i(n)}{\lambda ^i}. \end{aligned}$$

(59)

Then, the spectral problems $L(n) \bigl (\psi (\lambda ,n)\bigr ) = \lambda \psi (\lambda ,n)$ for $\psi =\psi _A$ and for $\psi =\psi _B$ recast into the following equations:

$$\begin{aligned}&\lambda \, e^{y(\lambda ,n)} \; + \; f(n) \,-\, \lambda \; + \; g(n) \, \lambda ^{-1} e^{-y(\lambda ,n-1)} \; = \; 0, \end{aligned}$$

(60)

$$\begin{aligned}&\lambda \, e^{-z(\lambda ,n-1)} \; + \; f(n) \,-\, \lambda \; + \; g(n+1) \, \lambda ^{-1} e^{z(\lambda ,n)} \; = \; 0, \end{aligned}$$

(61)

yielding recursions of the form (as equivalent conditions to (60)–(61))

$$\begin{aligned} y_{k+1}(n)&= - \sum _{\begin{array}{c} m_1,\ldots ,m_k\ge 0\\ \sum _{i=1}^{k} im_i=k+1 \end{array}} \frac{\prod _{i=1}^k y_i(n)^{m_i}}{\prod _{i=1}^k m_i!} -f(n)\delta _{k,0} \nonumber \\&\quad -g(n) \sum _{\begin{array}{c} m_1,\ldots ,m_{k-1}\ge 0\\ \sum _{i=1}^{k-1} im_i=k-1 \end{array}} \frac{\prod _{i=1}^{k-1} (-1)^{m_i} y_i(n-1)^{m_i}}{\prod _{i=1}^{k-1} m_i!}, \end{aligned}$$

(62)

$$\begin{aligned} z_{k+1}(n)&= \sum _{\begin{array}{c} m_1,\ldots ,m_k\ge 0\\ \sum _{i=1}^{k} im_i=k+1 \end{array}} \frac{\prod _{i=1}^k (-1)^{m_i} z_i(n)^{m_i}}{\prod _{i=1}^k m_i!} \; + \; f(n+1)\delta _{k,0} \nonumber \\&\quad + g(n+2) \sum _{\begin{array}{c} m_1,\ldots ,m_{k-1}\ge 0\\ \sum _{i=1}^{k-1} im_i=k-1 \end{array}} \frac{\prod _{i=1}^{k-1} z_i(n+1)^{m_i}}{\prod _{i=1}^{k-1} m_i!}, \end{aligned}$$

(63)

where $k\ge 0$. From these recursions, it easily follows that $y_k,z_k\in V$, $k\ge 0$. This proves the existence of wave functions of type A and of type B meeting the definitions in Sect. 1.3. Clearly, $\psi _A$ and $\psi _B$ are unique up to multiplying by arbitrary series $G(\lambda )$ and $E(\lambda )$ of $\lambda ^{-1}$ with constant coefficient of the form $G(\lambda )\in 1+ {{\mathbb {C}}}\left[ \left[ \lambda ^{-1}\right] \right] \lambda ^{-1}$ and $E(\lambda ) \in 1+ {{\mathbb {C}}}\left[ \left[ \lambda ^{-1}\right] \right] \lambda ^{-1}$. Since $\psi _A(\lambda ,n)=\bigl (1+{\mathrm{O}}(\lambda ^{-1})\bigr ) \,\lambda ^n$ and since $\psi _B(\lambda ,n)=\left( 1+{\mathrm{O}}\left( \lambda ^{-1}\right) \right) \, e^{-s(n)} \lambda ^{-n}$, we find that the $d(\lambda ,n)$ defined in (27) must have the form

$$\begin{aligned} d(\lambda ,n)\; = \; \lambda \, e^{-s(n-1)} \, e^{\sum _{k\ge 1} d_k(n) \, \lambda ^{-k}}. \end{aligned}$$

Then, by using the definitions of wave functions and of s(n), one easily derives that

$$\begin{aligned} e^{s(n)} \, d(\lambda ,n+1)\; = \; e^{s(n-1)} \, d(\lambda ,n). \end{aligned}$$

(64)

It follows that all $d_k(n)$, $k\ge 1$, are constants. Therefore, for any fixed choice of $\psi _A$, we can suitably choose the factor $E(\lambda )$ for $\psi _B$ such that $\psi _A,\psi _B$ form a pair. This proves the existence of pair of wave functions associated with f(n), g(n).

We proceed with the time dependence. Let $(v(n,\mathbf{t}),w(n,\mathbf{t}))$ be the unique solution in $V[[\mathbf{t}]]^2$ to the Toda lattice hierarchy satisfying the initial condition $v(n,\mathbf{0})=f(n)$, $w(n,\mathbf{0})=g(n)$. Let $L(n,\mathbf{t}) := \Lambda +v(n,\mathbf{t}) + w(n,\mathbf{t}) \, \Lambda ^{-1}$. Define $\sigma (n,\mathbf{t})$ as the unique up to a constant function satisfying the following equations:

$$\begin{aligned}&w(n,\mathbf{t}) \; = \; e^{\sigma (n-1,\mathbf{t})-\sigma (n,\mathbf{t})}, \end{aligned}$$

(65)

$$\begin{aligned}&\frac{\partial \sigma (n,\mathbf{t})}{\partial t_p} \; = \; - S_p (n,\mathbf{t}),\quad p\ge 0. \end{aligned}$$

(66)

An element $\psi _A(n,\mathbf{t},\lambda )= \bigl (1+\mathrm{O}\bigl (\lambda ^{-1}\bigr )\bigr ) \, \lambda ^n \, e^{\sum _{k\ge 0} t_k\lambda ^{k+1}}$ in ${\widetilde{V}}\left[ \left[ \mathbf{t}, \lambda ^{-1}\right] \right] \, \lambda ^n e^{\sum _{k\ge 0} t_k\lambda ^{k+1}}$ is called a wave function of type A associated with $(v(n,\mathbf{t}),w(n,\mathbf{t}))$ if

$$\begin{aligned}&L(n,\mathbf{t}) \, \bigl (\psi _A(\lambda ,n,\mathbf{t})\bigr ) \; = \; \lambda \, \psi _A(\lambda ,n,\mathbf{t}) , \quad \frac{\partial \psi _A}{\partial t_k} \; = \; \bigl (L^{k+1}\bigr )_+ \bigl (\psi _A\bigr ). \end{aligned}$$

(67)

An element $\psi _B(n,\mathbf{t},\lambda ) = \bigl (1+\mathrm{O}\bigl (\lambda ^{-1}\bigr )\bigr ) \lambda ^{-n} e^{-\sum _{k\ge 0} t_k\lambda ^{k+1}}$ in ${\widetilde{V}}\left[ \left[ \mathbf{t},\lambda ^{-1}\right] \right] e^{-\sigma (n,\mathbf{t})} \lambda ^{-n} e^{-\sum _{k\ge 0} t_k\lambda ^{k+1}}$ is called a wave function of type B associated with $(v(n,\mathbf{t}),w(n,\mathbf{t}))$ if

$$\begin{aligned}&L(n,\mathbf{t}) \, \bigl (\psi _B(\lambda ,n,\mathbf{t})\bigr ) \; = \; \lambda \, \psi _B(\lambda ,n,\mathbf{t}) , \quad \frac{\partial \psi _B}{\partial t_k} \; = \; - \bigl (L^{k+1}\bigr )_- \bigl (\psi _B\bigr ). \end{aligned}$$

(68)

The existence of wave functions $\psi _A$ and $\psi _B$ of type A and of type B associated with $(v(n,\mathbf{t}),w(n,\mathbf{t}))$ is a standard result in the theory of integrable systems (cf. [5, 6, 13, 27]); therefore, we omit its details. Denote

$$\begin{aligned} d(\lambda ,n,\mathbf{t})\, := \, \psi _A(\lambda ,n,\mathbf{t}) \, \psi _B(\lambda ,n-1,\mathbf{t}) \,-\, \psi _B(\lambda ,n,\mathbf{t}) \, \psi _A(\lambda ,n-1,\mathbf{t}), \end{aligned}$$

(69)

and introduce

$$\begin{aligned} m(\mu , \lambda , n,\mathbf{t}) \, := \, \frac{R(\mu ,n,\mathbf{t})}{\mu -\lambda } \; + \; Q(\mu ,n,\mathbf{t}), \end{aligned}$$

(70)

where $Q(\mu ,n,\mathbf{t}) := -\frac{{\mathrm{id}}}{\mu } \; + \; \begin{pmatrix} 0 &{}\quad 0 \\ 0 &{}\quad \gamma (\mu ,n,\mathbf{t}) \\ \end{pmatrix}$. We know from, e.g., [10] that the wave function $\psi _A(\lambda ,n,\mathbf{t})$ satisfies

$$\begin{aligned}&\nabla (\mu ) \, \begin{pmatrix} \psi _A(\lambda ,n,\mathbf{t}) \\ \psi _A(\lambda ,n-1,\mathbf{t}) \end{pmatrix} \; = \; m(\mu ,\lambda ,n,\mathbf{t}) \, \begin{pmatrix} \psi _A(\lambda ,n,\mathbf{t}) \\ \psi _A(\lambda ,n-1,\mathbf{t}) \end{pmatrix}. \end{aligned}$$

(71)

Similarly, the wave function $\psi _B(\lambda ,n,\mathbf{t})$ satisfies

$$\begin{aligned}&\nabla (\mu ) \, \begin{pmatrix} \psi _B(\lambda ,n,\mathbf{t}) \\ \psi _B(\lambda ,n-1,\mathbf{t}) \end{pmatrix} \; = \; \biggl (m(\mu ,\lambda ,n,\mathbf{t}) \, - \, \frac{\lambda }{\mu (\mu -\lambda )} I \biggr )\, \begin{pmatrix} \psi _B(\lambda ,n,\mathbf{t}) \\ \psi _B(\lambda ,n-1,\mathbf{t}) \end{pmatrix}. \end{aligned}$$

(72)

Here, I denotes the $2\times 2$ identity matrix.

Lemma 4

The following formula holds true:

$$\begin{aligned} \nabla (\mu ) \, \bigl (d(\lambda ,n,\mathbf{t})\bigr ) \; = \; \biggl ( - \frac{1}{\mu } + \gamma (\mu ,n,\mathbf{t}) \biggr ) \, d(\lambda ,n,\mathbf{t}). \end{aligned}$$

(73)

Proof

Recalling definition (69) for d and using (71)–(72), we find

$$\begin{aligned} \nabla (\mu ) \, \bigl ( d(\lambda ,n,\mathbf{t}) \bigr )&\; = \; \biggl ( {\mathrm{tr}}\bigl (m(\mu ,\lambda ,n,\mathbf{t})\bigr ) - \frac{\lambda }{\mu (\mu -\lambda )} \biggr ) \, d(\lambda ,n,\mathbf{t}). \end{aligned}$$

(74)

The lemma is then proved via a straightforward computation. $\square$

Definition 1

We say $\psi _A,\psi _B$ form a pair if $e^{\sigma (n-1,\mathbf{t})} d(\lambda ,n,\mathbf{t})=\lambda$.

The next lemma shows the existence of a pair.

Lemma 5

There exist a pair of wave functions $\psi _A,\psi _B$ associated with $(v(n,\mathbf{t}),w(n,\mathbf{t}))$. Moreover, the freedom of the pair is characterized by a factor $G(\lambda )$ via

$$\begin{aligned}&\displaystyle \psi _A(\lambda ,n,\mathbf{t}) \; \mapsto \; G(\lambda ) \, \psi _A(\lambda ,n,\mathbf{t}), \quad \psi _B(\lambda ,n,\mathbf{t}) \; \mapsto \; \frac{1}{G(\lambda )} \, \psi _B(\lambda ,n,\mathbf{t}), \end{aligned}$$

(75)

$$\begin{aligned}&\displaystyle G(\lambda ) \; = \; \sum _{j\ge 0} G_j \lambda ^{-j}, \quad G_0\; = \;1 \end{aligned}$$

(76)

with $G_j$, $j\ge 1$ being arbitrary constants.

Proof

Firstly, the freedom of a wave function $\psi _A$ associated with (v, w) is characterized by the multiplication by a factor $G(\lambda )$ of the form (76). Fix an arbitrary choice of $\psi _A$. For $\psi _B$ being a wave function of type B associated with (v, w), from (69) and the definitions of wave functions, we know $e^{\sigma (n-1,\mathbf{t})}d(\lambda ,n,\mathbf{t})$ must have the form

$$\begin{aligned} e^{\sigma (n-1,\mathbf{t})}d(\lambda ,n,\mathbf{t}) \; = \; \lambda \, e^{\sum _{k\ge 1} d_k(n,\mathbf{t}) \, \lambda ^{-k}} \end{aligned}$$

(77)

for some $d_k(n,\mathbf{t})$, $k\ge 1$. By using (67), (68), (69), we find

$$\begin{aligned}&d(\lambda ,n+1,\mathbf{t}) \; = \; w(n,\mathbf{t}) \, d(\lambda ,n,\mathbf{t}) \; = \; e^{\sigma (n-1,\mathbf{t})-\sigma (n,\mathbf{t})} \, d(\lambda ,n,\mathbf{t}) , \end{aligned}$$

i.e.,

$$\begin{aligned} e^{\sigma (n,\mathbf{t})} d(\lambda ,n+1,\mathbf{t})\; = \; e^{\sigma (n-1,\mathbf{t})} \, d(\lambda ,n,\mathbf{t}) , \end{aligned}$$

(78)

Using Lemma 4 and (66), we have

$$\begin{aligned}&\nabla (\mu ) \Bigl (e^{\sigma (n-1,\mathbf{t})}d(\lambda ,n,\mathbf{t})\Bigr ) \\&\quad \; = \; e^{\sigma (n-1,\mathbf{t})} \nabla (\mu ) \bigl (\sigma (n-1,\mathbf{t})\bigr ) d(\lambda ,n,\mathbf{t}) \; + \; e^{\sigma (n-1,\mathbf{t})} \nabla (\mu ) \bigl (d(\lambda ,n,\mathbf{t})\bigr ) \\&\quad \; = \; - e^{\sigma (n-1,\mathbf{t})}\sum _{p\ge 0} \frac{S_p(n-1,\mathbf{t})}{\mu ^{p+2}} d(\lambda ,n,\mathbf{t}) \\&\qquad \qquad + e^{\sigma (n-1,\mathbf{t})} d(\lambda ,n,\mathbf{t}) \biggl ( - \frac{1}{\mu } \; + \; \gamma (\mu ,n,\mathbf{t}) \biggr ) \; = \; 0. \end{aligned}$$

So we have

$$\begin{aligned} \frac{\partial (e^{\sigma (n-1,\mathbf{t})}d(\lambda ,n,\mathbf{t}))}{\partial t_p} \; = \; 0, \quad \forall \, p\ge 0. \end{aligned}$$

(79)

We deduce from (77), (78), (79) that $d_k(n,\mathbf{t})$, $k\ge 1$, are all constants. Therefore, there exists a unique choice of $\psi _B$ such that $\psi _A,\psi _B$ form a pair. The lemma is proved. $\square$

4 The k-point generating series

Let $(v,w)=(v(n,\mathbf{t}),w(n,\mathbf{t}))\in V[[\mathbf{t}]]^2$ be the unique solution to the Toda lattice hierarchy with the initial value $(v(n,\mathbf{0}),w(n,\mathbf{0}))=(f(n),g(n))$, and $(\psi _A,\psi _B)$ a pair of wave functions associated with (v, w). Define

$$\begin{aligned} \Psi _{{\mathrm{pair}}}(\lambda ,n,\mathbf{t}) \; = \; \begin{pmatrix} \psi _A (\lambda ,n,\mathbf{t}) &{}\quad \psi _B(\lambda ,n,\mathbf{t}) \\ \psi _A(\lambda ,n-1,\mathbf{t}) &{}\quad \psi _B(\lambda ,n-1,\mathbf{t}) \end{pmatrix}. \end{aligned}$$

(80)

Proposition 3

The following identity holds true:

$$\begin{aligned} R(\lambda , n, \mathbf{t}) \; \equiv \; \Psi _{{\mathrm{pair}}}(\lambda ,n,\mathbf{t}) \, \begin{pmatrix} 1 &{} \quad 0 \\ 0 &{} \quad 0 \end{pmatrix} \, \Psi _{{\mathrm{pair}}}^{-1}(\lambda ,n,\mathbf{t}) . \end{aligned}$$

(81)

Proof

Define

$$\begin{aligned}M\; = \;M(\lambda ,n,\mathbf{t})\, := \, \Psi _{{\mathrm{pair}}}(\lambda ,n,\mathbf{t}) \, \begin{pmatrix} 1 &{} \quad 0 \\ 0 &{}\quad 0 \end{pmatrix} \, \Psi _{{\mathrm{pair}}}^{-1}(\lambda ,n,\mathbf{t}). \end{aligned}$$

It is easy to verify that M satisfies

$$\begin{aligned} \bigl [{\mathcal {L}},M\bigr ] \, \bigl (\Psi _{{\mathrm{pair}}}\bigr ) \; = \; 0 , \qquad \det \, M \; = \; 0. \end{aligned}$$

The entries of M in terms of the pair of wave functions read

$$\begin{aligned} M=\frac{1}{d(\lambda ,n,\mathbf{t})} \begin{pmatrix} \psi _A(\lambda ,n,\mathbf{t}) \, \psi _B(\lambda ,n-1,\mathbf{t}) &{}\quad -\psi _A(\lambda ,n,\mathbf{t}) \, \psi _B(\lambda ,n,\mathbf{t}) \\ \psi _A(\lambda ,n-1,\mathbf{t}) \, \psi _B(\lambda ,n-1,\mathbf{t}) &{}\quad -\psi _A(\lambda ,n-1,\mathbf{t}) \, \psi _B(\lambda ,n,\mathbf{t}) \end{pmatrix} , \end{aligned}$$

(82)

where we recall that $d(\lambda ,n,\mathbf{t}) = \psi _A(\lambda ,n,\mathbf{t}) \, \psi _B(\lambda ,n-1,\mathbf{t}) - \psi _B(\lambda ,n,\mathbf{t}) \, \psi _A(\lambda ,n-1,\mathbf{t})$, which coincides with the determinant of $\Psi (\lambda ,n,\mathbf{t})$. It follows from $\psi _A(\lambda ,n,\mathbf{t})= \bigl (1+\mathrm{O}\bigl (\lambda ^{-1}\bigr )\bigr ) \, \lambda ^n \, e^{\sum _{k\ge 0} t_k\lambda ^{k+1}}$ and $\psi _B(\lambda ,n,\mathbf{t}) = \bigl (1+\mathrm{O}\bigl (\lambda ^{-1}\bigr )\bigr ) \, e^{-\sigma (n,\mathbf{t})} \lambda ^{-n} \, e^{-\sum _{k\ge 0} t_k\lambda ^{k+1}}$ that

$$\begin{aligned} M(\lambda ) \,-\, \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix} \; \in \; {\mathrm{Mat}}\left( 2,{\widetilde{V}}\left[ \left[ \mathbf{t},\lambda ^{-1}\right] \right] \lambda ^{-1}\right) . \end{aligned}$$

(83)

The proposition then follows from the uniqueness theorem proved in Sect. 2. $\square$

Define

$$\begin{aligned} D(\lambda ,\mu ,n,\mathbf{t}) \, := \, \frac{\psi _A(\lambda ,n,\mathbf{t}) \, \psi _B(\mu ,n-1,\mathbf{t}) \,-\, \psi _A(\lambda ,n-1,\mathbf{t}) \,\psi _B(\mu ,n,\mathbf{t})}{\lambda -\mu }.\nonumber \\ \end{aligned}$$

(84)

Theorem 2

Fix $k\ge 2$ being an integer. The generating series of k-point correlation functions of the solution $(v(n,\mathbf{t}),w(n,\mathbf{t}))$ has the following expression:

$$\begin{aligned}&\sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}(n,\mathbf{t})}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}} \nonumber \\&\quad \; = \; (-1)^{k-1} \frac{e^{k \sigma (n-1,\mathbf{t})}}{\prod _{j=1}^k \lambda _{j}} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k D(\lambda _{\pi (j)},\lambda _{\pi (j+1)},n,\mathbf{t}) \,-\, \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2} . \end{aligned}$$

(85)

Proof

It follows from (81) that

$$\begin{aligned} R(\lambda , n, \mathbf{t}) \; = \; \frac{r_1(\lambda ,n,\mathbf{t})^T r_2(\lambda ,n,\mathbf{t})}{d(\lambda ,n,\mathbf{t})} , \end{aligned}$$

(86)

where $r_1(\lambda ,n,\mathbf{t}):=(\psi _A(\lambda ,n,\mathbf{t}), \psi _A(\lambda ,n-1,\mathbf{t}))$, $r_2(\lambda ,n,\mathbf{t}):=(\psi _B(\lambda ,n-1,\mathbf{t}),-\psi _B(\lambda ,n,\mathbf{t}))$. Substituting this expression into the identity

$$\begin{aligned}&\sum _{i_1,i_2\ge 0} \frac{\Omega _{i_1,i_2}(n,\mathbf{t})}{\lambda _1^{i_1+2} \lambda _2^{i_2+2}} \; = \; \frac{{\mathrm{Tr}}\, \bigl (R_1(\lambda _1,n,\mathbf{t}) \, R_2(\lambda _2,n,\mathbf{t})\bigr )}{(\lambda _1-\lambda _2)^2} \,-\, \frac{1}{(\lambda _1-\lambda _2)^2}, \end{aligned}$$

(87)

we obtain

$$\begin{aligned} \sum _{i_1,i_2\ge 0} \frac{\Omega _{i_1,i_2}(n,\mathbf{t})}{\lambda _1^{i_1+2} \lambda _2^{i_2+2}}&\; = \; \frac{{\mathrm{Tr}}\, \bigl (r_1(\lambda _1,n,\mathbf{t})^T r_2(\lambda _1,n,\mathbf{t}) \, r_1(\lambda _2,n,\mathbf{t})^T r_2(\lambda _2,n,\mathbf{t})\bigr )}{d(\lambda _1,n,\mathbf{t}) d(\lambda _2,n,\mathbf{t}) (\lambda _1-\lambda _2)^2}\nonumber \\&\qquad \,-\, \frac{1}{(\lambda _1-\lambda _2)^2} \nonumber \\&\; = \; \frac{\bigl (r_2(\lambda _2,n,\mathbf{t}) \, r_1(\lambda _1,n,\mathbf{t})^T \bigr ) \, \bigl (r_2(\lambda _1,n,\mathbf{t}) \, r_1(\lambda _2,n,\mathbf{t})^T \bigr )}{ d(\lambda _1,n,\mathbf{t}) d(\lambda _2,n,\mathbf{t}) (\lambda _1-\lambda _2)^2}\nonumber \\&\qquad \,-\, \frac{1}{ (\lambda _1-\lambda _2)^2} \nonumber \\&\; = \; - \frac{D(\lambda _1,\lambda _2,n,\mathbf{t}) \, D(\lambda _2,\lambda _1,n,\mathbf{t}) }{ \lambda _1 \lambda _2 \, e^{-2\sigma (n-1,\mathbf{t})} } \,-\, \frac{1}{(\lambda _1-\lambda _2)^2}, \end{aligned}$$

(88)

where we used definition (84) and

$$\begin{aligned}&\frac{\psi _A(\lambda ,n,\mathbf{t}) \, \psi _B(\mu ,n-1,\mathbf{t}) \,-\, \psi _A(\lambda ,n-1,\mathbf{t}) \,\psi _B(\mu ,n,\mathbf{t})}{\lambda -\mu } \nonumber \\&\quad \; = \; \frac{r_2(\mu ,n,\mathbf{t}) \, r_1(\lambda ,n,\mathbf{t})^T}{\lambda -\mu }. \end{aligned}$$

This proves the $k=2$ case of (85). For $k\ge 3$, the proof is similar. Indeed,

$$\begin{aligned}&\sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}(n,\mathbf{t})}{\lambda _1^{i_1+1} \ldots \lambda _k^{i_k+1}} \nonumber \\&\qquad \; = \; - \sum _{\pi \in {\mathcal {S}}_k/C_k} \frac{{\mathrm{Tr}}\, \Bigl (\prod _{j=1}^k r_1\bigl (\lambda _{\pi (j)},n,\mathbf{t}\bigr )^T r_2\bigl (\lambda _{\pi (j)},n,\mathbf{t}\bigr ) \Bigr )}{e^{-k \, \sigma (n-1,\mathbf{t})} \, \prod _{j=1}^k \bigl (\lambda _{\pi {(j)}}-\lambda _{\pi {(j+1)}}\bigr )} \nonumber \\&\qquad \; = \; - \sum _{\pi \in {\mathcal {S}}_k/C_k} \frac{r_2\bigl (\lambda _{\pi (k)},n,\mathbf{t}\bigr ) \, r_1\bigl (\lambda _{\pi (1)},n,\mathbf{t}\bigr )^T \, \ldots \, r_2\bigl (\lambda _{\pi (k-1)},n,\mathbf{t}\bigr ) \, r_1\bigl (\lambda _{\pi (k)},n,\mathbf{t}\bigr )^T}{e^{-k \, \sigma (n-1,\mathbf{t})} \, \prod _{j=1}^k \bigl (\lambda _{\pi {(j)}}-\lambda _{\pi {(j+1)}}\bigr )} \nonumber \\&\qquad \; = \; - \frac{(-1)^k}{e^{-k \, \sigma (n-1,\mathbf{t})}} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k D\bigl (\lambda _{\pi (j)}, \lambda _{\pi (j+1)},n,\mathbf{t}\bigr ). \end{aligned}$$

(89)

This proves the $k\ge 3$ case of (85). The theorem is proved. $\square$

Remark 2

In (85) or (30), the freedom (75) affects the $D(\lambda , \mu )$ through multiplying it by a factor of the form $\frac{G(\lambda )}{G(\mu )}$, but the product $\prod _{j=1}^k D(\lambda _{\pi (j)},\lambda _{\pi (j+1)})$ remains unchanged.

In Appendix A, the abstract form of (85) is obtained, where a pair of abstract pre-wave functions are introduced.

Proof of Theorem 1

Taking $\mathbf{t}=\mathbf{0}$ on the both sides of (85) gives (30). $\square$

Write

$$\begin{aligned} \psi _A(\lambda ,n,\mathbf{t}) \; = \; \phi _A(\lambda ,n,\mathbf{t}) \, \lambda ^n, \qquad \psi _B(\lambda ,n,\mathbf{t}) \; = \; \phi _B(\lambda ,n,\mathbf{t}) \, e^{-\sigma (n,\mathbf{t})} \, \lambda ^{-n}.\quad \end{aligned}$$

(90)

Theorem 1 can then be alternatively written in terms of $\phi _A,\phi _B$ by the following corollary.

Corollary 1

The following formula holds true for $k\ge 2$:

$$\begin{aligned} \sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}(n,\mathbf{t})}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}}&= (-1)^{k-1} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k B(\lambda _{\pi (j)},\lambda _{\pi (j+1)},n,\mathbf{t})- \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2} , \end{aligned}$$

(91)

where $B(\lambda ,\mu ,n,\mathbf{t})$ is defined by

$$\begin{aligned} B(\lambda ,\mu ,n,\mathbf{t}) := \frac{\phi _A(\lambda ,n,\mathbf{t}) \, \phi _B(\mu ,n-1,\mathbf{t}) \,-\, w(n,\mathbf{t})\, \phi _A(\lambda ,n-1,\mathbf{t}) \,\phi _B(\mu ,n,\mathbf{t})}{\lambda -\mu }.\nonumber \\ \end{aligned}$$

(92)

In particular, let $\phi _A(\lambda ,n):=e^{(\Lambda -1)^{-1}(y(\lambda ,n))}$, $\phi _B(\lambda ,n):=e^{(\Lambda -1)^{-1}(z(\lambda ,n))} e^{-s(n)}$ (cf. (60)–(61)), and let $B(\lambda ,\mu ,n):= \frac{\phi _A(\lambda ,n) \, \phi _B(\mu ,n-1) \,-\, g(n)\, \phi _A(\lambda ,n-1) \,\phi _B(\mu ,n)}{\lambda -\mu }$, then we have

$$\begin{aligned} \sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}(n,\mathbf{0})}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}}&= (-1)^{k-1} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k B(\lambda _{\pi (j)},\lambda _{\pi (j+1)},n) \nonumber \\&\quad -\, \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2} . \end{aligned}$$

(93)

For some particular examples related to matrix models, it turns out that the suitable chosen D coincides, possibly up to simple factors, with certain kernel of the matrix model. However, the D is not unique. We now introduce a formal series $K(\lambda ,\mu )$ such that the generating series of multi-point correlation functions still has an explicit expression, but this time K is local and is therefore unique for the given solution. The series K is defined by

$$\begin{aligned} K(\lambda ,\mu ) \, := \, \frac{(1\; + \;\alpha (\lambda ))(1\; + \;\alpha (\mu )) - w_0 \, \gamma (\lambda ) \, \Lambda \bigl (\gamma (\mu )\bigr )}{\lambda -\mu }, \end{aligned}$$

(94)

where $1+\alpha (\lambda )$ is the (1,1) entry of the basic matrix resolvent $R(\lambda )$, and $\gamma (\lambda )$ is the (2,1) entry. The next theorem expresses the left-hand side of (85) in terms of K.

Theorem 3

For any $k\ge 2$, the following formula holds true:

$$\begin{aligned} \sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}}&\; = \; (-1)^{k-1} \frac{\sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k K\bigl (\lambda _{\pi (j)},\lambda _{\pi (j+1)}\bigr )}{ \prod _{i=1}^k \bigl (1\; + \;\alpha (\lambda _i)\bigr )} \nonumber \\&\quad \qquad \quad -\, \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2} . \end{aligned}$$

(95)

Proof

The identity (81) gives

$$\begin{aligned}&\psi _B(\lambda ,n-1,\mathbf{t}) \; = \; \frac{(1\; + \;\alpha (\lambda ,n,\mathbf{t})) \, d(\lambda ,n,\mathbf{t})}{\psi _A(\lambda ,n,\mathbf{t})}, \\&\psi _B(\lambda ,n,\mathbf{t}) \; = \; - \, \frac{\beta (\lambda ,n,\mathbf{t}) \, d(\lambda ,n,\mathbf{t})}{\psi _A(\lambda ,n,\mathbf{t})} \; = \; w_n\,\frac{\gamma (\lambda ,n+1,\mathbf{t}) \, d(\lambda ,n,\mathbf{t})}{\psi _A(\lambda ,n,\mathbf{t})}, \\&\psi _A(\lambda ,n-1,\mathbf{t}) \; = \; \psi _A(\lambda ,n,\mathbf{t}) \, \frac{\gamma (\lambda ,n,\mathbf{t})}{1\; + \;\alpha (\lambda ,n,\mathbf{t})}. \end{aligned}$$

Substituting these expressions into (84), we obtain

$$\begin{aligned} D(\lambda ,\mu ,n,\mathbf{t}) \; = \; d(\mu ,n,\mathbf{t}) \frac{\psi _A(\lambda ,n,\mathbf{t})}{\psi _A(\mu ,n,\mathbf{t})} e(\lambda ,\mu ,n,\mathbf{t}), \end{aligned}$$

(96)

where

$$\begin{aligned} e(\lambda ,\mu ,n,\mathbf{t}){:=} \frac{(1\; + \;\alpha (\lambda ,n,\mathbf{t}))(1\; + \;\alpha (\mu ,n,\mathbf{t})) - w_n(\mathbf{t}) \, \gamma (\lambda ,n,\mathbf{t}) \, \gamma (\mu ,n+1,\mathbf{t}) }{(\lambda -\mu ) \, (1\; + \;\alpha (\lambda ,n,\mathbf{t}))}.\nonumber \\ \end{aligned}$$

(97)

Combining with the definition of $K(\lambda ,\mu ,n,\mathbf{t})$ and Theorem 1, we find

$$\begin{aligned}&\sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}(n,\mathbf{t})}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}} \nonumber \\&\quad \; = \; (-1)^{k-1} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k K \bigl (\lambda _{\pi (j)},\lambda _{\pi (j+1)},n,\mathbf{t}\bigr ) \,-\, \frac{ \delta _{k,2}}{(\lambda _1-\lambda _2)^2}. \end{aligned}$$

(98)

The theorem is proved. $\square$

It seems to be an interesting question to study the geometric and algebraic meaning of the kernel K (as well as D). Below we give without proof some of their properties.

Proposition 4

The functions K and D are related to

$$\begin{aligned}&K(\lambda ,\mu ,n,\mathbf{t}) \; = \;\frac{e^{\sigma (n-1,\mathbf{t})} }{\mu } \, \bigl (1+\alpha (\lambda ,n,\mathbf{t})\bigr ) \, \frac{\psi _A(\mu ,n,\mathbf{t})}{\psi _A(\lambda ,n,\mathbf{t})} \, D(\lambda ,\mu ,n,\mathbf{t}) \\&\quad \quad \quad \quad \quad \; = \;\frac{e^{2\sigma (n-1,\mathbf{t})}}{\lambda \,\mu } \, \psi _A(\mu ,n,\mathbf{t}) \, \psi _B(\lambda ,n-1,\mathbf{t})\, D(\lambda ,\mu ,n,\mathbf{t}) \\&\quad \quad \quad \quad \quad \; = \;\frac{e^{\sigma (n-1,\mathbf{t})}}{\lambda } \, \bigl (1+\alpha (\mu ,n,\mathbf{t})\bigr )\,\frac{\psi _B(\lambda ,n-1,\mathbf{t}) }{\psi _B(\mu ,n-1,\mathbf{t})} \, D(\lambda ,\mu ,n,\mathbf{t}). \end{aligned}$$

We observe that the following three formal series

$$\begin{aligned} K(\lambda ,\mu )-\frac{1+\alpha (\lambda )}{\lambda -\mu }, \quad K(\lambda ,\mu )-\frac{1+\alpha (\mu )}{\lambda -\mu }, \quad K(\lambda ,\mu )-\frac{2+\alpha (\lambda )+\alpha (\mu )}{2 \, (\lambda -\mu )} \end{aligned}$$

all belong to ${{\mathcal {A}}}\left[ \left[ \lambda ^{-1},\mu ^{-1}\right] \right]$. Therefore, the following three formal series

$$\begin{aligned}&\displaystyle K(\lambda ,\mu ,n,\mathbf{t})-\frac{1+\alpha (\lambda ,n,\mathbf{t})}{\lambda -\mu }, \quad K(\lambda ,\mu ,n,\mathbf{t})-\frac{1+\alpha (\mu ,n,\mathbf{t})}{\lambda -\mu }, \\&\displaystyle K(\lambda ,\mu ,n,\mathbf{t})-\frac{2+\alpha (\lambda ,n,\mathbf{t})+\alpha (\mu ,n,\mathbf{t})}{2(\lambda -\mu )} \end{aligned}$$

all belong to $V[[\mathbf{t}]]\left[ \left[ \lambda ^{-1},\mu ^{-1}\right] \right]$. It follows from this observation and Proposition 4 that

$$\begin{aligned} \frac{e^{s(n-1)} }{\mu }\, D(\lambda ,\mu ,n,\mathbf{0}) \, \biggl (\frac{\mu }{\lambda }\biggr )^n \,-\, \frac{1}{\lambda -\mu } \; \in \; {\widetilde{V}}\left[ \left[ \lambda ^{-1},\mu ^{-1}\right] \right] . \end{aligned}$$

(99)

Remark 3

We could loosen both the conditions for wave functions and the pair condition. Let us say $\psi _A$ and $\psi _B$ are pre-wave functions of type A and of type B, respectively, if they satisfy the first equations of (67) and (68). Define $d_{{\mathrm{pre}}}(\lambda ,n,\mathbf{t})$ and $D_{{\mathrm{pre}}}(\lambda ,\mu ,n,\mathbf{t})$ by (129) and (140). Then, the following formula holds true:

$$\begin{aligned}&\sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}(n,\mathbf{t})}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}} \nonumber \\&\quad \qquad \; = \; \frac{(-1)^{k-1}}{\prod _{j=1}^k d_{{\mathrm{pre}}}(\lambda _j,n,\mathbf{t})} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k D_{{\mathrm{pre}}}(\lambda _{\pi (j)},\lambda _{\pi (j+1)},n,\mathbf{t}) \,-\, \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2} . \end{aligned}$$

(100)

Now, $\psi _A$ and $\psi _B$ are determined by $(v(n,\mathbf{t}),w(n,\mathbf{t}) )$ up to

$$\begin{aligned} \psi _A (\lambda ,n,\mathbf{t}) \; \mapsto \; G(\lambda ,\mathbf{t}) \, \psi _A(\lambda ,n,\mathbf{t}), \quad \psi _B (\lambda ,n,\mathbf{t}) \; \mapsto \; E(\lambda ,\mathbf{t}) \, \psi _B (\lambda ,n,\mathbf{t}), \end{aligned}$$

where $G(\lambda ,\mathbf{t})=1+\sum _{k\ge 1} G_k(\mathbf{t}) \lambda ^{-k}$, $E(\lambda ,\mathbf{t})=1+\sum _{k\ge 1} E_k(\mathbf{t})\lambda ^{-k}$ with $G_k(\mathbf{t}), E_k(\mathbf{t}) \in {{\mathbb {C}}}[[\mathbf{t}]]$, $k\ge 1$. This freedom affects $D_{{\mathrm{pre}}}(\lambda ,\mu ,n,\mathbf{t})$ and $d_{{\mathrm{pre}}}(\lambda ,n,\mathbf{t})$ into

$$\begin{aligned} D_{{\mathrm{pre}}}(\lambda ,\mu ,n,\mathbf{t})\mapsto & {} G(\lambda ,\mathbf{t}) \, E(\mu ,\mathbf{t}) \, D_{{\mathrm{pre}}}(\lambda ,\mu ,\mathbf{t}) , \quad \nonumber \\ d_{{\mathrm{pre}}}(\lambda ,n,\mathbf{t})\mapsto & {} G(\lambda ,\mathbf{t}) \, E(\lambda ,\mathbf{t}) \, d_{{\mathrm{pre}}}(\lambda ,\mathbf{t}). \end{aligned}$$

Therefore, it gives rise to each summand of (100) the factor

$$\begin{aligned} \frac{\prod _{j=1}^k G(\lambda _{\pi (j)},\mathbf{t}) E(\lambda _{\pi (j+1)},\mathbf{t})}{\prod _{j=1}^k G(\lambda _j,\mathbf{t}) E(\lambda _j,\mathbf{t})}, \end{aligned}$$

which is equal to 1. Hence, the right-hand side of (100) remains unchanged.

5 Applications

Partition functions in some matrix models and enumerative models are particular tau-functions for the Toda lattice hierarchy. Theorem 1 can then be used for computing their logarithmic derivatives. In this section, we do two explicit computations.

5.1 Application I: enumeration of ribbon graphs

The initial data of the GUE solution to the Toda lattice hierarchy are given by $f(n)=0$ and $g(n)=n$; see, for example, [10] for the proof. For this case, we can take $V={{\mathbb {Q}}}[n]$ and ${\widetilde{V}}=V$. Substituting the initial data in (26), we find

$$\begin{aligned} s(n)&\; = \; - \bigl (1-\Lambda ^{-1}\bigr )^{-1} \log g(n) \nonumber \\&\; = \; - \bigl (1-\Lambda ^{-1}\bigr )^{-1} \log n\; = \;-\log \,\Gamma (n+1) \; + \;C, \end{aligned}$$

(101)

where C is a constant. Below, we fix this constant as 0.

Proposition 5

The $\psi _A,\psi _B$ defined by

$$\begin{aligned}&\psi _A(\lambda ,n)= \sum _{j\ge 0} (-1)^j \frac{(n-2j+1)_{2j}}{2^j\, j!\, \lambda ^{2j}} \lambda ^n, \end{aligned}$$

(102)

$$\begin{aligned}&\psi _B(\lambda ,n)= \Gamma (n+1) \sum _{j\ge 0} \frac{(n+1)_{2j}}{2^j\,j!\, \lambda ^{2j}} \lambda ^{-n} \end{aligned}$$

(103)

form a particular pair of wave functions associated with $(f(n)=0,g(n)=n)$. Here and below $(a)_{i}$ denotes the increasing Pochhammer symbol defined by $(a)_{i}=a(a+1)\ldots (a+i-1)$.

Proof

It is straightforward to verify that both $\psi _A$ and $\psi _B$ satisfy the equation

$$\begin{aligned} \psi (\lambda ,n+1) \; + \; n\, \psi (\lambda ,n-1) \; = \; \lambda \, \psi (\lambda ,n). \end{aligned}$$

(104)

Moreover, from definitions (102)–(103), we see that

$$\begin{aligned} \psi _A \in {\widetilde{V}}\bigl ( \bigl (\lambda ^{-1}\bigr ) \bigr )\, \lambda ^n, \quad \psi _B \in {\widetilde{V}}\bigl ( \bigl (\lambda ^{-1}\bigr ) \bigr )e^{-s(n)} \lambda ^{-n}. \end{aligned}$$

We are left to show that

$$\begin{aligned} \Gamma (n)^{-1} \, \Bigl (\psi _A(\lambda ,n) \, \psi _B(\lambda ,n-1) - \psi _B(\lambda ,n) \, \psi _A(\lambda ,n-1) \Bigr ) \; = \; \lambda . \end{aligned}$$

(105)

Clearly, the meaning of this identity is the following: Both sides of (105) are Laurent series of $\lambda ^{-1}$ with coefficients in ${\widetilde{V}}=V=Q[n]$, and the equality means all the coefficients should be equal. More precisely, the identity (105) can be equivalently written as the following sequence of identities:

$$\begin{aligned}&\frac{n}{j+1}\sum _{j_1=0}^{j+1} \frac{(-1)^{j_1}}{2} \left( {\begin{array}{c}j+1\\ j_1\end{array}}\right) \left( {\begin{array}{c}n+2j_1-1\\ 2j+1\end{array}}\right) \nonumber \\&\quad + \sum _{j_1=0}^j (-1)^{j_1} \left( {\begin{array}{c}j\\ j_1\end{array}}\right) \left( {\begin{array}{c}n+2j_1\\ 2j+1\end{array}}\right) \; = \; 0,\quad j\ge 0. \end{aligned}$$

(106)

From (64), we know that the left-hand side of (106) as a polynomial of n is a constant for any $j\ge 0$. Note that the value of the left-hand side of (106) at $n=0$ is obviously 0 for any $j\ge 0$. The proposition is proved. $\square$

It follows from the above proposition an explicit expression for the $D(\lambda ,\mu ,n,\mathbf{0})$ (cf. Eq. (84)) associated with the pair (102)–(103):

$$\begin{aligned} \frac{e^{s(n-1)}}{\mu }\, D(\lambda ,\mu ,n,\mathbf{0}) \, \biggl (\frac{\mu }{\lambda }\biggr )^n \; = \; \frac{1}{\lambda -\mu } \; + \; A(\lambda ,\mu ,n), \end{aligned}$$

(107)

with $A(\lambda ,\mu ,n)$ given by

$$\begin{aligned} A(\lambda ,\mu ,n)&= \sum _{k\ge 1} \frac{(2k-1)!!}{(2k)!}\sum _{p=0}^{2k-1} (-1)^{p+[(p+1)/2]} \left( \begin{array}{c} k-1 \\ {[p/2]} \end{array}\right) \nonumber \\&\quad \qquad \cdot \prod _{j=-p}^{2k-1-p} (n+j) \, \lambda ^{-p-1} \mu ^{-(2k-p)} . \end{aligned}$$

(108)

This explicit expression (108) first appeared in [31]. Denote

$$\begin{aligned} {\widehat{A}}(\lambda ,\mu ,n)\; = \;\frac{1}{\lambda -\mu } \; + \; A(\lambda ,\mu ,n). \end{aligned}$$

(109)

As a corollary of Proposition 5, Theorem 1, and the above (107), we have achieved a new proof of the following theorem of Jian Zhou.

Theorem 4

[31] Fix $k\ge 2$ being an integer. The generating series of k-point connected GUE correlators has the following expression:

$$\begin{aligned} \sum _{i_1,\ldots ,i_k\ge 1} \frac{\bigl \langle {\mathrm{tr}} \, M^{i_1}\ldots {\mathrm{tr}} \, M^{i_k}\bigr \rangle _{{\mathrm{c}}}}{\lambda _1^{i_1+1} \ldots \lambda _k^{i_k+1}}&\; = \; (-1)^{k-1} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k {\widehat{A}} \, \bigl (\lambda _{\pi (j)},\lambda _{\pi (j+1)},n\bigr ) \nonumber \\&\qquad \quad \,-\, \frac{ \delta _{k,2}}{(\lambda _1-\lambda _2)^2}, \end{aligned}$$

(110)

where ${\widehat{A}}$ is defined by (108)–(109). Here, we recall that for any fixed $i_1,\ldots ,i_k$, the connected GUE correlator $\langle {\mathrm{tr}} \, M^{i_1}\ldots {\mathrm{tr}} \, M^{i_k}\rangle _{{\mathrm{c}}}$ is a polynomial of n (cf. [4, 10, 17, 21]).

5.2 Application II: Gromov–Witten invariants of ${\mathbb {P}}^1$ in the stationary sector

The initial data for the Gromov–Witten solution to the Toda lattice hierarchy were, for example, derived in [10,11,12]. It has the following explicit expression:

$$\begin{aligned} f(n)\; = \;n\epsilon +\frac{\epsilon }{2}, \quad g(n)\; = \;1. \end{aligned}$$

(111)

We have

$$\begin{aligned} s(n) \; = \; - \bigl (1-\Lambda ^{-1}\bigr )^{-1} \log 1 \; = \; C, \end{aligned}$$

where C is an arbitrary constant. Below, we take $C=0$.

Proposition 6

The $\psi _1,\psi _2$ defined by

$$\begin{aligned}&\psi _A(\lambda ,n) \; = \; \epsilon ^{\frac{\lambda }{\epsilon }-\frac{1}{2}} \, \Gamma \Bigl (\frac{\lambda }{\epsilon }+\frac{1}{2}\Bigr )\, J_{\frac{\lambda }{\epsilon }-n-\frac{1}{2}}\Bigl (\frac{2}{\epsilon }\Bigr ), \end{aligned}$$

(112)

$$\begin{aligned}&\psi _B(\lambda ,n) \; = \; (-1)^{n+1} \epsilon ^{-\frac{\lambda }{\epsilon }-\frac{1}{2}} \, \lambda \, \Gamma \Bigl (-\frac{\lambda }{\epsilon }+\frac{1}{2}\Bigr )\, J_{-\frac{\lambda }{\epsilon }+n+\frac{1}{2}}\Bigl (\frac{2}{\epsilon }\Bigr ) \end{aligned}$$

(113)

form a particular pair of wave functions associated with $f(n)=n\epsilon +\frac{\epsilon }{2},g(n)=1$. Here, $J_\nu (y)$ denotes the Bessel function, and the right-hand sides of (112)–(113) are understood as the large $\lambda$ asymptotics of the corresponding analytic functions.

Proof

Firstly, using the properties of Bessel functions, we can verify that $\psi _A(\lambda ,n)$ and $\psi _B(\lambda ,n)$ defined from the above asymptotics satisfy

$$\begin{aligned}&\psi _A(\lambda ,n+1)\; + \; \Bigl (n\epsilon +\frac{\epsilon }{2}\Bigr ) \, \psi _A(\lambda ,n) \; + \; \psi _A(\lambda ,n-1) \; = \; \lambda \, \psi _A(\lambda ,n),\\&\psi _B(\lambda ,n+1)\; + \; \Bigl (n\epsilon +\frac{\epsilon }{2}\Bigr ) \, \psi _B(\lambda ,n) \; + \; \psi _B(\lambda ,n-1) \; = \; \lambda \, \psi _B(\lambda ,n). \end{aligned}$$

Secondly, as $\lambda$ goes to $\infty$, the following asymptotics hold true:

$$\begin{aligned}&\epsilon ^{\frac{\lambda }{\epsilon }-\frac{1}{2}} \, \Gamma \Bigl (\frac{\lambda }{\epsilon }+\frac{1}{2}\Bigr )\, J_{\frac{\lambda }{\epsilon }-n-\frac{1}{2}}\Bigl (\frac{2}{\epsilon }\Bigr ) \;\sim \; \lambda ^n \Bigl (1\; + \; {\mathrm{O}}\bigl (\lambda ^{-1}\bigr )\Bigr ),\\&(-1)^{n+1} \epsilon ^{-\frac{\lambda }{\epsilon }-\frac{1}{2}} \, \lambda \, \Gamma \Bigl (-\frac{\lambda }{\epsilon }+\frac{1}{2}\Bigr )\, J_{-\frac{\lambda }{\epsilon }+n+\frac{1}{2}}\Bigl (\frac{2}{\epsilon }\Bigr ) \;\sim \; \lambda ^{-n} \Bigl (1\; + \; {\mathrm{O}}\bigl (\lambda ^{-1}\bigr )\Bigr ). \end{aligned}$$

Thirdly, $\psi _A$ and $\psi _B$ also satisfy

$$\begin{aligned} \psi _A(\lambda ,n) \, \psi _B(\lambda ,n-1) \,-\, \psi _B(\lambda ,n) \, \psi _A(\lambda ,n-1) \; = \; \lambda . \end{aligned}$$

We have verified all the defining properties for a pair of wave functions associated with $f(n)=n\epsilon +\frac{\epsilon }{2},g(n)=1$. The proposition is proved. $\square$

Note that

$$\begin{aligned}&\psi _A(\lambda ,n-1)\; = \; \epsilon ^{\frac{\lambda }{\epsilon }-\frac{1}{2}} \, \Gamma \Bigl (\frac{\lambda }{\epsilon }+\frac{1}{2}\Bigr )\, J_{\frac{\lambda }{\epsilon }-n+\frac{1}{2}}\Bigl (\frac{2}{\epsilon }\Bigr ), \end{aligned}$$

(114)

$$\begin{aligned}&\psi _B(\lambda ,n-1)\; = \; (-1)^{n} \epsilon ^{-\frac{\lambda }{\epsilon }-\frac{1}{2}} \, \lambda \, \Gamma \Bigl (-\frac{\lambda }{\epsilon }+\frac{1}{2}\Bigr )\, J_{-\frac{\lambda }{\epsilon }+n-\frac{1}{2}}\Bigl (\frac{2}{\epsilon }\Bigr ), \end{aligned}$$

(115)

and denote

$$\begin{aligned} J_\nu (y) \,=:\, \frac{(y/2)^\nu }{\Gamma (\nu +1)} j_{\nu +\frac{1}{2}}(y^2/4). \end{aligned}$$

It follows from (84), (112)–(115)  that the $D(\lambda ,\mu ,0,\mathbf{0})$ associated with the pair (112)–(113) has the following explicit expression:

$$\begin{aligned} \frac{1}{\mu }D(\lambda ,\mu ,0,\mathbf{0}) \; = \; - \frac{1}{\epsilon }\, \frac{j_{-\frac{\mu }{\epsilon }}\bigl (\frac{1}{\epsilon ^2}\bigr ) \, j_{\frac{\lambda }{\epsilon }}\bigl (\frac{1}{\epsilon ^2}\bigr ) + \frac{\epsilon ^{-2}}{(\frac{1}{2}-\frac{\mu }{\epsilon })(\frac{1}{2}+\frac{\lambda }{\epsilon })} \, j_{1-\frac{\mu }{\epsilon }}\bigl (\frac{1}{\epsilon ^2}\bigr )\, j_{1+\frac{\lambda }{\epsilon }}\bigl (\frac{1}{\epsilon ^2}\bigr )}{\mu /\epsilon -\lambda /\epsilon }. \end{aligned}$$

Then, according to [12], the function $\frac{1}{\mu }D(\lambda ,\mu ,0,\mathbf{0})$ has the following expressions:

$$\begin{aligned}&\frac{1}{\mu }D(\lambda ,\mu ,0,\mathbf{0}) \nonumber \\&\quad \; = \; -\frac{1}{\epsilon }\sum _{k=0}^\infty \frac{(a-b-2k+1)_{k-1}}{k! \, (-a+\tfrac{1}{2})_k \, (b+\tfrac{1}{2})_k} \epsilon ^{-2k} \end{aligned}$$

(116)

$$\begin{aligned}&\quad \; = \; \frac{-1}{\epsilon (a-b)} ~ {}_2 F_3\Bigl (\frac{b-a}{2}, \frac{b-a+1}{2}; \, \frac{1}{2} -a \,, \, \frac{1}{2} + b, b-a+1; \, -4 \epsilon ^{-2}\Bigr ) \end{aligned}$$

(117)

$$\begin{aligned}&\quad \ \sim \; \frac{-1}{\epsilon (a-b)} \,-\, \sum _{p,q\ge 0} \frac{(-1)^{q+1}}{a^{p+1} b^{q+1}} \sum _{k\ge 1} \frac{\epsilon ^{-2k-1}}{k!} \nonumber \\&\quad \qquad \quad \sum _{1\le i,j\le k} (-1)^{i+j}\frac{(i+j-2k)_{k-1} \bigl (i-\frac{1}{2}\bigr )^p \bigl (j-\frac{1}{2}\bigr )^q}{(i-1)!(j-1)!(k-i)!(k-j)!} \;=:\; {\widehat{A}} \, (\lambda ,\mu ), \end{aligned}$$

(118)

where $a:=\frac{\mu }{\epsilon }$, $b:=\frac{\lambda }{\epsilon }$, the $(a-b+1)_{-1}$ of (116) is defined as $1/(a-b)$, and $\sim$ in (118) is taken as $a,b\rightarrow \infty$ away from the half integers. The explicit expression (118) first appeared in [12]. So we have completed a new proof of the following theorem.

Theorem 5

[12] The generating series of k-point ($k\ge 2$) Gromov–Witten invariants of ${\mathbb {P}}^1$ in the stationary sector has the following explicit expression:

$$\begin{aligned}&\epsilon ^k \! \sum _{i_1,\ldots ,i_k\ge 0} \frac{(i_1+1)! \ldots (i_k+1)!}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}} \langle \tau _{i_1}(\omega ) \ldots \tau _{i_k}(\omega ) \rangle (\epsilon ) \nonumber \\&\quad \; = \; (-1)^{k-1} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{i=1}^k {\widehat{A}} \, \bigl (\lambda _{\pi (i)}, \lambda _{\pi (i+1)}\bigr ) \,-\, \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2}, \end{aligned}$$

(119)

where ${\widehat{A}} \, (\lambda ,\mu )$ is explicitly defined in (118), and

$$\begin{aligned} \langle \tau _{i_1}(\omega ) \ldots \tau _{i_k}(\omega ) \rangle (\epsilon ) := \sum _{g\ge 0} \epsilon ^{2g-2} \sum _{d\ge 0} \int _{\left[ \overline{{\mathcal {M}}}_{g,k}({\mathbb {P}}^1,d)\right] ^{{\mathrm{virt}}}} {\mathrm{ev}}_1^*(\omega ) \ldots {\mathrm{ev}}_k^*(\omega ) \, \psi _1^{i_1} \ldots \psi _k^{i_k}. \end{aligned}$$

(120)

(See, for example, [12] for the notation about the integral in the right-hand side of (120).)

Appendix A: Pair of abstract pre-wave functions

Here, we construct a ring that is suitable for defining abstract pre-wave functions. Recall that ${{\mathcal {A}}}$ is the ring of polynomials of $v_k,w_k$, $k \in {{\mathbb {Z}}}$. Instead of the ${{\mathbb {Z}}}$-coefficients, we will use in this appendix the ${{\mathbb {Q}}}$-coefficients, i.e., ${{\mathcal {A}}}={{\mathbb {Q}}}\bigl [ \{v_k,w_k \,|\, k\in {{\mathbb {Z}}}\}\bigr ]$, is now under consideration. For each monic monomial $\alpha \in {{\mathcal {A}}}\backslash {{\mathbb {Q}}}$, we associate a symbol $m_\alpha$. Denote by ${\mathcal {B}}$ the polynomial ring

$$\begin{aligned} {\mathcal {B}}\, := \, {{\mathbb {Q}}}\bigl [\{ \,m_\alpha \,|\,\alpha \text{ is } \text{ a } \text{ monic } \text{ monomial } \text{ in } {{\mathcal {A}}}\backslash {{\mathbb {Q}}}\} \bigr ]. \end{aligned}$$

(121)

Define the action of $\Lambda ^k$ on ${\mathcal {B}}$ with $k\in {{\mathbb {Z}}}$ by

$$\begin{aligned} \Lambda ^k \bigl (m_{\alpha _1} \ldots m_{\alpha _l}) \; = \; m_{ \Lambda ^k(\alpha _1)} \ldots m_{ \Lambda ^k (\alpha _l)} \end{aligned}$$

(122)

for $\alpha _1,\ldots ,\alpha _l$ being monic monomials in ${{\mathcal {A}}}\backslash {{\mathbb {Q}}}$, as well as by linearly extending it to other elements of ${\mathcal {B}}$. For a monic monomial $\alpha =v_{i_1} \ldots v_{i_r} w_{j_1} \ldots w_{j_s}\in {{\mathcal {A}}}\backslash {{\mathbb {Q}}}$ with $i_1\le \cdots \le i_r$, $j_1\le \cdots \le j_s$ and $r+s\ge 1$, let $k_\alpha :=-i_1$ (if $r\ge 1$), $k_\alpha :=-j_1$ (if $r=0$); the monomial $\Lambda ^{k_\alpha } (\alpha )\in {{\mathcal {A}}}$ is then called the (unique) reduced monomial (associated to $\alpha$). Denote by ${\mathcal {C}}$ the polynomial ring generated by $m_{\beta }$, $v_k$, $w_k$ with ${{\mathbb {Q}}}$-coefficients, where $\beta$ are reduced monic monomials, and $k\in {{\mathbb {Z}}}$. Let us also define an action of $\Lambda ^k$ on ${\mathcal {C}}$, $k\in {{\mathbb {Z}}}$. To this end, we introduce some notations: For $\beta$ a reduced monic monomial of ${{\mathcal {A}}}$, denote

$$\begin{aligned} n_{\Lambda ^k(\beta )} \, := \, \left\{ \begin{array}{cc} m_\beta + \sum _{i=0}^{k-1} \Lambda ^i(\beta ), &{} ~ k\ge 0,\\ m_\beta - \sum _{i=k}^{-1} \Lambda ^i(\beta ), &{} ~ k\le -1. \end{array}\right. \end{aligned}$$

(123)

Then, for a monomial $\alpha \cdot m_{\beta _1} \ldots m_{\beta _s}$ of ${\mathcal {C}}$ with $\alpha$ being a monomial in ${{\mathcal {A}}}$, define

$$\begin{aligned} \Lambda ^k (\alpha \cdot m_{\beta _1} \ldots m_{\beta _s}) = \Lambda ^k (\alpha ) \cdot \prod _{j=1}^s n_{\Lambda ^k(\beta _j)}, \quad k\in {{\mathbb {Z}}}. \end{aligned}$$

(124)

Define the action of $\Lambda ^k$ on other elements in ${\mathcal {C}}$ by requiring it as a linear operator. Denote by $p:{\mathcal {B}}\rightarrow {\mathcal {C}}$ the linear map which maps $m_{\alpha _1} \ldots m_{\alpha _l}\in {\mathcal {B}}$ to $n_{\alpha _1} \ldots n_{\alpha _l}\in {\mathcal {C}}$, for $\alpha _i$, $i=1,\ldots ,l$ being monic monomials in ${{\mathcal {A}}}\backslash {{\mathbb {Q}}}$. Denote by ${\mathcal {B}}^0$ the image of p. Clearly, ${{\mathcal {A}}}\subset {\mathcal {B}}^0$. Indeed, the element $(\Lambda -1) \bigl (\sum _{i=1}^l \lambda _i \, m_{\alpha _i}\bigr ) \in {\mathcal {B}}$ becomes $\sum _{i=1}^l \lambda _i \alpha _i \in {{\mathcal {A}}}$ under the map p. Here, $\alpha _1,\ldots ,\alpha _l$ are distinct monic monomials in ${{\mathcal {A}}}\backslash {{\mathbb {Q}}}$. Finally, we define an operator ${\mathbb {S}}: {{\mathcal {A}}}\backslash {{\mathbb {Q}}}\rightarrow {\mathcal {B}}^0$ by

$$\begin{aligned} {\mathbb {S}} \bigl (\lambda _1 \alpha _1 \; + \; \cdots \; + \; \lambda _l \alpha _l\bigr ) \; = \; \lambda _1 n_{\alpha _1} \; + \; \cdots \; + \; \lambda _l n_{\alpha _l} \, \end{aligned}$$

(125)

for $\alpha _1,\ldots ,\alpha _l$ being distinct monic monomials and $\lambda _1,\ldots ,\lambda _l\in {{\mathbb {Q}}}$.

Motivated by (62) and (63), define two families of elements $y_i,z_i \in {{\mathcal {A}}}$, $i\ge 1$ by

$$\begin{aligned}&y_{k+1} \ = - \sum _{\begin{array}{c} m_1,\ldots ,m_k\ge 0\\ \sum _{i=1}^{k} im_i=k+1 \end{array}} \frac{\prod _{i=1}^k y_i^{m_i}}{\prod _{i=1}^k m_i!} -v_0\delta _{k,0} \\&\qquad \qquad \quad - w_0 \sum _{\begin{array}{c} m_1,\ldots ,m_{k-1}\ge 0 \sum _{i=1}^{k-1} im_i=k-1 \end{array}} \frac{\prod _{i=1}^{k-1} (-1)^{m_i} \bigl (\Lambda ^{-1}(y_i)\bigr )^{m_i}}{\prod _{i=1}^{k-1} m_i!}, \\&z_{k+1} \; = \; \sum _{\begin{array}{c} m_1,\ldots ,m_k\ge 0\\ \sum _{i=1}^{k} im_i=k+1 \end{array}} \frac{\prod _{i=1}^k (-1)^{m_i} z_i^{m_i}}{\prod _{i=1}^k m_i!} \; + \; v_1\delta _{k,0} \\&\quad \qquad \qquad + w_2 \sum _{\begin{array}{c} m_1,\ldots ,m_{k-1}\ge 0\\ \sum _{i=1}^{k-1} im_i=k-1 \end{array} }\frac{\prod _{i=1}^{k-1} \bigl (\Lambda (z_i)\bigr )^{m_i}}{\prod _{i=1}^{k-1} m_i!}. \end{aligned}$$

Equivalently, the generating series $y(\lambda ):=\sum _{i\ge 1} y_i/\lambda ^i$, $z(\lambda ):=\sum _{i\ge 1} z_i/\lambda ^i$ satisfy

$$\begin{aligned}&\lambda \, e^{y(\lambda )} \; + \; v_0 \,-\, \lambda \; + \; w_0 \, \lambda ^{-1} \Lambda ^{-1}\bigl (e^{-y(\lambda )}\bigr ) \; = \; 0, \\&\lambda \, \Lambda ^{-1} \bigl (e^{-z(\lambda )}\bigr ) \; + \; v_0 \,-\, \lambda \; + \; w_1 \, \lambda ^{-1} e^{z(\lambda )} \; = \; 0. \end{aligned}$$

Define

$$\begin{aligned} \psi _A\, := \, e^{{\mathbb {S}} (y(\lambda ))} \otimes \lambda ^n \otimes 1, \quad \psi _B \, := \, e^{{\mathbb {S}} (z(\lambda ))} \otimes \lambda ^{-n} \otimes e^{-\sigma } , \end{aligned}$$

(126)

where $e^{-\sigma }$ is a formal element satisfying $e^{(1-\Lambda ^{-1})(-\sigma )} = w_0$, and $\lambda ^n$, $\lambda ^{-n}$ are formal elements satisfying $\Lambda ^k (1\otimes \lambda ^n)=\lambda ^k \otimes \lambda ^n$, $\Lambda ^k (1\otimes \lambda ^{-n})=\lambda ^{-k} \otimes \lambda ^{-n}$, $k\in {{\mathbb {Z}}}$. We have

$$\begin{aligned}&L \bigl (\psi _A\bigr ) \; = \; \lambda \, \psi _A,\quad L \bigl (\psi _B\bigr ) \; = \; \lambda \, \psi _B,\nonumber \\&\psi _A(\lambda ) \; = \; \bigl (1+{\mathrm{O}}\bigl (\lambda ^{-1}\bigr )\bigr )\otimes \lambda ^n \;\in \; {\mathcal {C}} \left[ \left[ \lambda ^{-1}\right] \right] \otimes \lambda ^n, \end{aligned}$$

(127)

$$\begin{aligned}&\psi _B(\lambda ) \; = \; \bigl (1+{\mathrm{O}}\bigl (\lambda ^{-1}\bigr )\bigr ) \otimes \lambda ^{-n} \otimes e^{-\sigma } \;\in \; {\mathcal {C}} \left[ \left[ \lambda ^{-1}\right] \right] \otimes \lambda ^{-n} \otimes e^{-\sigma }, \end{aligned}$$

(128)

where $L=\Lambda +v_0 + w_0 \, \Lambda ^{-1}$. We call $\psi _A$ and $\psi _B$ the abstract pre-wave functions of type A and of type B, respectively, associated with $v_0,w_0$.

Denote

$$\begin{aligned} d_{{\mathrm{pre}}}(\lambda )\, := \, \psi _A(\lambda ) \, \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) \,-\, \psi _B(\lambda ) \, \Lambda ^{-1}\bigl (\psi _A(\lambda )\bigr ) \end{aligned}$$

(129)

and

$$\begin{aligned} \Psi (\lambda ) \, := \, \begin{pmatrix} \psi _A (\lambda ) &{}\quad \psi _B(\lambda ) \\ \Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) &{}\quad \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) \end{pmatrix}. \end{aligned}$$

(130)

Then, we have the following identity:

$$\begin{aligned} R(\lambda ) \; = \; \Psi (\lambda ) \, \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix} \, \Psi ^{-1}(\lambda ) \;=:\; M(\lambda ). \end{aligned}$$

(131)

The proof is similar to that of Proposition 3. (The main fact used in the proof is that from the definition, the coefficients of entries of $R(\lambda )$ are uniquely determined in an algebraic way.) We omit its details here. However, let us explain the equality (131) by an equivalent form. From definition, we have

$$\begin{aligned} M(\lambda ) \; = \; \frac{1}{d_{{\mathrm{pre}}}(\lambda )} \begin{pmatrix} \psi _A (\lambda ) \, \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) &{}\quad -\psi _A (\lambda ) \, \psi _B(\lambda ) \\ \Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) \, \Lambda ^{-1} \bigl (\psi _B(\lambda )\bigr ) &{}\quad -\Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) \, \psi _B(\lambda ) \end{pmatrix}. \end{aligned}$$

Then, from a straightforward calculation by using the definitions, we find

$$\begin{aligned} M_{11}(\lambda )&\; = \; \frac{1}{1 \,-\, \frac{w_0}{\lambda ^2} \, e^{ \Lambda ^{-1} (z(\lambda ) - y(\lambda ))}}, \end{aligned}$$

(132)

$$\begin{aligned} M_{12}(\lambda )&\; = \; \frac{1}{\lambda ^{-1} \, e^{-\Lambda ^{-1}(y(\lambda ))} \,-\, \frac{\lambda }{w_0} \, e^{-\Lambda ^{-1}(z(\lambda ))}}, \end{aligned}$$

(133)

$$\begin{aligned} M_{21}(\lambda )&\; = \; \frac{1}{\lambda \, e^{\Lambda ^{-1}(y(\lambda ))} \,-\, \frac{w_0}{\lambda } \, e^{\Lambda ^{-1}(z(\lambda ))}}, \end{aligned}$$

(134)

$$\begin{aligned} M_{22}(\lambda )&\; = \; \frac{1}{1 \,-\, \frac{\lambda ^2}{w_0} \, e^{ \Lambda ^{-1} (y(\lambda )- z(\lambda ))}}. \end{aligned}$$

(135)

Hence, the equality (131) means new expressions for the entries of the basic matrix resolvent $R(\lambda )$ explicitly in terms of $y(\lambda ),z(\lambda )$. Substituting the following expansions

$$\begin{aligned} y(\lambda ) \; = \; -\frac{v_0}{\lambda } \,-\, \frac{\frac{1}{2} v_0^2+w_0}{\lambda ^2} \; + \; \cdots , \qquad z(\lambda ) \; = \; \frac{v_1}{\lambda } \; + \; \frac{\frac{1}{2} v_1^2+w_2}{\lambda ^2} \; + \; \cdots \nonumber \\ \end{aligned}$$

(136)

into (132)–(135), we find that the new expressions agree with (24). Combining with (56), (57), we obtain

$$\begin{aligned}&\frac{1}{\frac{\lambda ^2}{w_0} \, e^{ \Lambda ^{-1} (y(\lambda )- z(\lambda ))} \,-\, 1} \; = \; \sum _{p\ge 0} \Omega _{p,0} \, \lambda ^{-p-2} \;=:\; A, \end{aligned}$$

(137)

$$\begin{aligned}&\frac{1}{\lambda \, e^{\Lambda ^{-1}(y(\lambda ))} \,-\, \frac{w_0}{\lambda } \, e^{\Lambda ^{-1}(z(\lambda ))}} \; = \; \lambda ^{-1}\; + \;\sum _{p\ge 0} \Lambda ^{-1} \bigl (S_p\bigr ) \, \lambda ^{-p-2} \; =: \; B. \end{aligned}$$

(138)

We therefore arrive at the following formulae:

$$\begin{aligned} e^{\Lambda ^{-1}(y(\lambda ))} \; = \; \frac{1}{\lambda }\, \frac{1+A}{B}, \qquad e^{\Lambda ^{-1}(z(\lambda ))} \; = \; \frac{\lambda }{w_0} \, \frac{A}{B}. \end{aligned}$$

(139)

Let us proceed to the generating series of multi-point correlation functions. Define

$$\begin{aligned} D_{{\mathrm{pre}}}(\lambda ,\mu ) \, := \, \frac{\psi _A(\lambda ) \, \Lambda ^{-1} \bigl (\psi _B(\mu )\bigr ) \,-\, \Lambda ^{-1} \bigl (\psi _A(\lambda )\bigr ) \,\psi _B(\mu )}{\lambda -\mu }. \end{aligned}$$

(140)

Using (131), Proposition 1, and a similar argument to the proof of Theorem 2, we obtain

$$\begin{aligned} \sum _{i_1,\ldots ,i_k\ge 0} \frac{\Omega _{i_1,\ldots ,i_k}}{\lambda _1^{i_1+2} \ldots \lambda _k^{i_k+2}}&= \frac{(-1)^{k-1}}{\prod _{j=1}^k d_{{\mathrm{pre}}}(\lambda _j)} \sum _{\pi \in {\mathcal {S}}_k/C_k} \prod _{j=1}^k D_{{\mathrm{pre}}}(\lambda _{\pi (j)},\lambda _{\pi (j+1)}) \nonumber \\&\quad -\, \frac{\delta _{k,2}}{(\lambda _1-\lambda _2)^2} . \end{aligned}$$

(141)

For the reader’s convenience, we give the first few terms of the abstract pre-wave functions $\psi _A(\lambda )$ and $\psi _B(\lambda )$ as follows:

$$\begin{aligned} \psi _A&= \biggl (1 \,-\, \frac{m_{v_0}}{\lambda } \; + \; \frac{m_{v_0}^2-m_{v_0^2}-2m_{w_0}}{2 \lambda ^2} \nonumber \\&\quad - \frac{1}{6\lambda ^3} \Bigl (m_{v_0}^3 + 2m_{v_0^3} - 3m_{v_0} m_{v_0^2} + 6m_{v_0 w_0} + 6m_{v_0 w_1} \nonumber \\&\quad - 6 m_{v_0} m_{w_0} - 6v_{-1} w_0\Bigr ) \; + \; {\mathrm{O}}\Bigl (\frac{1}{\lambda ^4}\Bigr )\biggr ) \, \lambda ^n,\end{aligned}$$

(142)

$$\begin{aligned} \psi _B&= \biggl (1 \; + \;\frac{m_{v_0}+v_0}{\lambda } \; + \; \frac{m_{v_0}^2+m_{v_0^2} +2v_0 m_{v_0}+2m_{w_0}+2v_0^2+2w_0+2w_1}{2\lambda ^2} \nonumber \\&\quad + \frac{1}{6 \lambda ^3} \Bigl (m_{v_0}^3+6m_{v_0} m_{w_0} +3 m_{v_0} m_{v_0^2} +2m_{v_0^3} +6m_{v_0 w_1} + 6m_{v_0 w_0} \nonumber \\&\quad +3 v_0 m_{v_0}^2+6v_0^2m_{v_0} +6w_0 m_{v_0} +6w_1 m_{v_0} +3v_0 m_{v_0^2}+6v_0 m_{w_0} \nonumber \\&\quad +6 v_0^3+12v_0 w_0+12 v_0 w_1 + 6 v_1 w_1 \Bigr ) \; + \; {\mathrm{O}}\Bigl (\frac{1}{\lambda ^4}\Bigr )\biggr ) \, \lambda ^{-n} e^{-\sigma }. \end{aligned}$$

(143)

It turns out that the above abstract pre-wave functions form a pair. Namely, $d_{{\mathrm{pre}}}(\lambda ) = \lambda \, e^{\Lambda ^{-1}(-\sigma )}$. Interestingly, for given arbitrary initial value (f(n), g(n)), based on this statement, one obtains a constructive method for a pair of wave functions associated with (f(n), g(n)) (cf. (28) in Sect. 1.3 for the definition of a pair). This is important considering Theorem 1. We hope to confirm the statement on the pair property of the abstract pre-wave functions in another publication.