The Spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions

Abstract

We consider the operator $Ly = - (d/dx)^{2}y + x^{2} y + w(x) y, \quad y \text { in} L^{2}(\mathbb {R}),$ where $w(x) = s \delta (x - b) + t \delta (x + b) , \quad b \neq 0 \, \, \text {real}, \quad s, t \in \mathbb {C}$. This operator has a discrete spectrum: eventually the eigenvalues are simple. Their asymptotic is given. In particular, if s=−t, $\lambda _{n} = (2n + 1) + s^{2}\, \frac {\kappa (n)}{n} + \rho (n) \label {eq:abstractlam}$ where $\kappa (n) = \frac {1}{2\pi } \left [(-1)^{n + 1} \sin \left (2 b \sqrt {2n} \right ) - \frac {1}{2} \sin \left (4 b \sqrt {2n} \right ) \right ]$ and $\vert \rho (n) \vert \leq C \frac {\log n}{n^{3/2}}. \label {eq:abstracterr}$ If $\overline {s} = -t$, the number T(s) of non-real eigenvalues is finite, and $T(s) \leq \left (C (1 + \vert s \vert ) \log (e + \vert s \vert ) \right )^{2}$. The analogue of the above asymptotic is given in the case of any two-point interaction perturbation.

1 Introduction

The operator

$$L^{0} = - \, \frac{d^{2}}{dx^{2}} + x^{2}, \quad x \in \mathbb{R}^{1}, $$

is the one-dimensional harmonic oscillator; this is an unbounded self-adjoint operator acting in $L^{2}(\mathbb {R})$. As one can see in any introductory book on quantum mechanics, L ⁰ has a discrete spectrum ${\Lambda }^{0} = \{z_{n}\}_{n = 0}^{\infty }$,

$$z_{n} = 2n + 1, \quad n = 0, 1, \dotsc $$

and a compact resolvent

$$ R^{0}(z) = (z - L^{0})^{-1} , \quad z \not\in {\Lambda}^{0}. $$

(1.1)

A normalized orthogonal system of eigenfunctions can be chosen as the Hermite functions

$$ h_{n}(x) = \left( \pi^{1/2} 2^{n} n! \right)^{-1/2} e^{-x^{2}/2} H_{n}(x), \quad n = 0, 1, \dotsc $$

(1.2)

where

$$ H_{n}(x) = e^{x^{2}/2} \left(e^{-x^{2}/2}\right)^{(n)} $$

(1.3)

are the Hermite polynomials.

Spectral analysis of perturbed operators

$$ L = L^{0} + W $$

(1.4)

with special W, in particular, the point interaction perturbations

$$ Wf = wf, \quad w(x) = {\sum}_{j = 1}^{J} c_{j} \delta \left( x - b_{j} \right), \quad J \text{ finite} $$

(1.5)

was studied in many mathematical and physical papers, for example [3, 4, 6–8, 20, 21, 24, 25].

In the series of papers [12–14] S. Fassari, F. Rinaldi and G. Inglese investigate the spectrum of L∈(1.4) when the perturbation

$$ W = - \tau \left( \delta \left( x - b \right) + \delta \left( x + b \right) \right), \quad \tau, b > 0, $$

(1.6)

i.e., L ⁰ is perturbed by a pair of attractive point interactions of equal strength whose centers are situated at the same distance from the origin. In this case the operator L=L ⁰+W is self-adjoint; the techniques used are based on Green’s function analysis.

E. Demiralp ([6–8]) found numerically the non-real eigenvalues of (1.6) when

$$ -\tau = i\gamma, \quad \gamma \text{ real} $$

for γ large enough.

H. Cartarius, D. Dast, D. Haag, G. Wunner, R. Eichler, and J. Main [5] and [16], motivated by analysis of Bose-Einstein condensates with $\mathcal {P}\mathcal {T}$-symmetric loss and gain, focused on the case of non-Hermitian perturbations

$$W = i \gamma \left[ \delta \left( x - b \right) - \delta \left( x + b \right) \right]. $$

Their numerical estimates showed that for small γ the spectrum of L=L ⁰+W is on the real line $\mathbb {R}$, and they gave some predictions on the state of decay of the disk radii where the eigenvalues of the operator L are located. Now we provide a rigorous mathematical analysis of the asymptotics of eigenvalues λ _n =λ _n (L ⁰+W).

We follow the techniques used in [2, 9–11, 19] and based on careful estimates related to the resolvent representation

$$ R = R^{0} + {\sum}_{j = 1}^{\infty} U_{j}, $$

(1.7)

$$ U_{0} = R^{0} , \quad U_{k} = R^{0} W U_{k - 1} = U_{k - 1} W R^{0}, \quad k \geq 0. $$

(1.8)

Moreover, we essentially use the property of perturbations W∈(1.5) to have such a matrix

$$ w_{jk} = \left\langle W h_{j}, h_{k} \right\rangle, \quad j, k = 0, 1, \dotsc $$

(1.9)

that for some α>0 there exists M>0 such that

$$ \vert w_{jk} \vert \leq \frac{M}{(1 + j)^{\alpha} (1 + k)^{\alpha}}, \quad j, k = 0, 1, \dotsc , $$

(1.10)

Detailed results on the spectrum and convergence of spectral decompositions of L=L ⁰+W for a general W under the condition (1.10) were given by B. Mityagin and P. Siegl in [19]. In the case (1.5) of the finite point interaction perturbations, $\displaystyle \alpha = \frac {1}{4}$.

2 Preliminaries, Technical Introduction, Review the Results

• 1 Our main concern is the harmonic oscillator operator (1.4) and its special perturbation W. We will focus on this case, although many constructions are very general and could be performed in analysis of other differential operators — see [1, 9, 19].

  Let L ⁰ be an operator in $\ell ^{2}(\mathbb {Z}_{+})$,

  $$ L^{0} e_{k} = z_{k} e_{k}, \quad z_{k} = (2k + 1), \quad k = 0, 1, \dotsc, $$

  (2.1)

  and $W = (w_{jk})_{0}^{\infty }$ a matrix such that for some α>0 and C ₀>0,

  $$ \vert w_{jk} \vert \leq \frac{C_{0}}{(1 + j)^{\alpha} (1 + k)^{\alpha}} $$

  (2.2)

  Then the KLMN Theorem [17, Chapter 6, §§1 – 4] leads to the definition of the closed operator

  $$ L = L^{0} + W $$

  (2.3)

  with a dense domain — see details in [19]. Let us recall some facts, introduce notations and explain a few elementary but important inequalities.

• 2 To adjust our constructions to the set of eigenvalues of the unperturbed operator (2.1), let us define strips

  $$\begin{array}{@{}rcl@{}} H_{n} & =& \left\lbrace z \in \mathbb{C}: \vert \Re z - z_{n} \vert \leq 1 \right\rbrace, \quad n \geq 1\\ H_{0} & = & \left\lbrace z \in \mathbb{C}: \Re z - z_{0} \leq 1 \right\rbrace \end{array} $$

  (2.4)

  and the squares

  $$ \mathcal{D}_{n} = \left\lbrace z \in H_{n}: \vert \Re z - z_{n} \vert \leq \frac{1}{2}, \vert \Im z \vert \leq \frac{1}{2} \right\rbrace, \quad n \geq 0 $$

  (2.5)

  around eigenvalues $\left\lbrace z_{n} \right\rbrace_{n =0}^{\infty} = \left\lbrace 2n + 1 \right\rbrace_{n = 0}^{\infty}$ in H _n .

  The resolvent

  $$ R(z) = (z - L^{0} - W)^{-1} $$

  (2.6)

  of the operator (2.3) is well-defined in the right half-plane

  $$ \left\lbrace z: \Re z \geq 2 N_{*} \right\rbrace \setminus \bigcup_{k = N_{*}}^{\infty} \mathcal{D}_{k} $$

  (2.7)

  outside of the disks $\mathcal {D}_{k}$, k≥N _∗, if N _∗ is large enough.

  It follows from the Neumann-Riesz decomposition

  $$ R = R^{0} + R^{0} W R^{0} + R^{0} W R^{0} W R^{0} + \dotsb = R^{0} + {\sum}_{j = 1}^{\infty} U_{j}, $$

  (2.8)

  where

  $$ U_{0} = R^{0} = (z - L^{0})^{-1}, \qquad U_{j} = R^{0} W U_{j - 1} = U_{j - 1} W R^{0}, \quad j \geq 1. $$

  (2.9)

  Of course, the convergence of the series should be explained at least in (2.7). This is done in [1, 19]; now I will remind only the estimates of N _∗ because it will be important later (see Theorem 4.4, (4.38) and Theorem 4.1, (4.11)) in accounting for points of the spectrum σ(L) outside of the real line.

• 3 Define a diagonal operator K,

  $$ K e_{j} = \frac{1}{\sqrt{z - z_{j}}} e_{j}, j = 0, 1, \dotsc, \Im z \neq 0 $$

  (2.10)

  with understanding that

  $$ \sqrt{\xi} = r^{1/2} e^{i \varphi / 2} \, \text{if} \, \xi = r e^{i \varphi}, \quad - \pi < \varphi \leq \pi. $$

  (2.11)

  Then K ²=R ⁰, $z \in \mathbb {C} \setminus \mathbb {R}$; maybe, we lose analyticity but rough estimates – when just the absolute values of matrix elements work well – are good enough.

  Indeed, (2.8), (2.9) could be rewritten as

  $$ R^{0} = K^{2}, \quad U_{j} = K (K W K)^{j} K, \quad R = R^{0} + {\sum}_{j = 1}^{\infty} K (K W K)^{j} K, $$

  (2.12)

  where

  $$ (K W K)_{k m} = \frac{1}{\sqrt{z - z_{k}}} W_{km} \frac{1}{\sqrt{z - z_{m}}},\quad k, m = 0, 1, 2, \dotsc, \quad z \in \mathbb{C} \setminus \mathbb{R}. $$

  (2.13)

Lemma 2.1

Under the assumptions ( 2.1 ), and ( 2.2 ), with $\displaystyle 0 < \alpha < \frac {1}{2}$ , if $z \in H_{n} \setminus \mathcal {D}_{n}$ , then KWK is a Hilbert-Schmidt operator, and

$$ \ell \equiv \Vert K W K \Vert_{\text{HS}} \leq \frac{C_{0} M(\alpha) \log (en)}{n^{2\alpha}} , \quad M(\alpha) \equiv 6 + \frac{4/3}{1- 2 \alpha} + \frac{1}{3\alpha} $$

(2.14)

Proof

If $z \in \partial \mathcal {D}_{n}$, i.e,

$$\begin{array}{@{}rcl@{}} z = (2n + 1) + \xi + i \eta , \quad \vert \xi \vert = \frac{1}{2}, \, \vert \eta \vert &\leq& \frac{1}{2},\\ \, \text{or} \quad \vert \xi \vert \leq \frac{1}{2}, \, \vert \eta \vert &=& \frac{1}{2}; \quad \xi, \eta \in \mathbb{R}, \end{array} $$

(2.15)

then

$$ \frac{1}{2} \leq \vert z - z_{j} \vert \leq 3, \, j = n, n \pm 1, $$

(2.16)

and if |n−j|≥2,

$$ \frac{3}{2} \vert n - j \vert \leq 2 \vert n - j \vert - 1 \leq \vert z - z_{j} \vert \leq 2 \vert n - j \vert + 1 \leq \frac{5}{2} \vert n - j \vert. $$

(2.17)

Therefore, by (2.2), (2.13),

$$ \ell^{2} = {\sum}_{j, k = 1}^{\infty} \frac{\vert w_{jk} \vert^{2}}{\vert z - z_{j} \vert \vert z - z_{k} \vert} \leq {C_{0}^{2}} \mu^{2} $$

(2.18)

with

$$ \mu = {\sum}_{j = 0}^{\infty} \frac{1}{(1 + j)^{2 \alpha} \vert z - z_{j} \vert}. $$

(2.19)

The sum of three terms for j=n,n±1 in (2.19) by (2.16) does not exceed

$$ 3 \cdot \frac{1}{n^{2\alpha}} \cdot 2 = \frac{6}{n^{2\alpha}}, $$

(2.20)

and by (2.17), the remaining part of μ, namely, $\displaystyle {\sum }_{j = 0}^{n - 2} + {\sum }_{j = n + 2}^{\infty }$, by the integral test does not exceed

$$\begin{array}{@{}rcl@{}} &&\frac{2}{3} \left[ \frac{1}{n} + \frac{1}{n^{2\alpha}} + {\int}_{0}^{n - 1} \frac{dx}{(1 + x)^{2\alpha} (n - x)} \right]\\ &&+ \frac{2}{3} \left[ \frac{1}{2} \cdot \left( \frac{1}{n + 3} \right)^{2\alpha} + {\int}_{n + 2}^{\infty} \frac{dy}{(1 + y)^{2\alpha} (y - n)} \right]. \end{array} $$

(2.21)

The first integral (after the change of variables x=n ξ) is

$$\begin{array}{@{}rcl@{}} \frac{1}{n^{2 \alpha}} {\int}_{0}^{1 - (1/n)} \frac{n \, d\xi}{\displaystyle n(1 - \xi) \left( \frac{1}{n} + \xi \right)^{2\alpha}} & \leq &\frac{1}{n^{2 \alpha}} \left[ 2 {\int}_{0}^{1/2} \frac{d\xi}{\xi^{2 \alpha}} + 2^{2\alpha} {\int}_{1/2}^{1 - (1/n)} \frac{d\xi}{1 - \xi} \right] =\\ & = &\left( \frac{2}{n} \right)^{2\alpha} \left[ \frac{1}{1-2\alpha} + \log \frac{n}{2} \right]. \end{array} $$

(2.22)

The second integral in (2.21) is equal to

$$\begin{array}{@{}rcl@{}} \frac{1}{n^{2\alpha}} {\int}_{1 + (2/n)}^{\infty} \frac{d\eta}{\displaystyle (\eta - 1)\left( \frac{1}{n} + \eta \right)^{2\alpha}} & \leq& \frac{1}{n^{2 \alpha}} \left[ {\int}_{1 + (2/n)}^{2} \frac{d\eta}{\eta - 1} + {\int}_{2}^{\infty} \frac{d\eta}{(\eta - 1)^{2\alpha + 1}} \right] =\\ & =& \frac{1}{n^{2\alpha}} \left[ \log \frac{n}{2} + \frac{1}{2\alpha} \right]. \end{array} $$

(2.23)

If we collect the inequalities (2.20) to (2.23), we get (with 2^2α≤2)

$$\begin{array}{@{}rcl@{}} \mu&\leq& \frac{2}{3} \frac{1}{n^{2\alpha}} \left[ 9 + \frac{3}{2} + \frac{2}{1 - 2\alpha} + 3 \log \frac{e n}{2} + \frac{1}{2\alpha} \right] \leq \\ &\leq& \frac{1}{n^{2\alpha}} [M(\alpha) + 2 \log n] \end{array} $$

(2.24)

$$ \text{where} \, M(\alpha) = 6 + \frac{4/3}{1- 2 \alpha} + \frac{1}{3\alpha}. $$

(2.25)

With (2.18) and (2.2) we come to (2.14). □

Of course, the constant factors in the inequalities (2.18) – (2.24) are not sharp but we get some idea on their magnitude. If $\alpha = \displaystyle \frac {1}{4}$ we have

$$ M\left( \frac{1}{4} \right) = 6 + \frac{4}{3} \cdot 2 + \frac{2}{3} < 10, \text{and} $$

(2.26)

$$ \mu \leq \frac{2}{\sqrt{n}} (5 + \log n) $$

(2.27)

This case is important in analysis of the harmonic oscillator and its perturbations (1.5). The estimates (2.26) and (2.27) will be used later as well.

Remark 2.2

Let $s \equiv \sum \limits _{\begin {array}{c}j = 0 \\ j \neq n\end {array}}^{\infty } \frac {1}{(1 + j)^{\beta }} \cdot \frac {1}{\vert n - j \vert }$. Then

$$\begin{array}{@{}rcl@{}} s \leq \frac{M(\beta)}{n^{\beta}} \log en, && if \, 0 < \beta \leq 1, \end{array} $$

(2.28.i)

$$\begin{array}{@{}rcl@{}} s \leq \frac{M}{n}, &&if \, \beta > 1. \end{array} $$

(2.28.ii)

Proof

The case β=2α<1 is done in the proof of Lemma 2.1. Other cases could be explained in the same way; we omit details. □

• 4 In this section we use properties of Shatten class operators and related equalities for the norms ∥T∥ _p , $T \in \mathfrak {S}_{p}$, of compact operators — see details in [15, 22]. By (2.10) the operator K is bounded if $z \in H_{n} \setminus \mathcal {D}_{n}$ and by (2.16), (2.17) its norm

  $$ \Vert K \Vert \leq \sqrt{2}. $$

  (2.29)

  Therefore, for U _j ∈(2.12) if j≥2

  $$ \Vert U_{j} \Vert_{1} \leq 2 \Vert K W K {\Vert_{2}^{j}} \leq 2 \ell^{j} \leq 2 \left[ M(\alpha) \frac{\log e n}{n^{2\alpha}} \right]^{j}. $$

  (2.30)

  But we can claim that U ₁ is a trace-class operator as well, and

  $$ \Vert U_{1} \Vert_{1} = \Vert K(KWK) K \Vert_{1} \leq \Vert K \Vert_{4} \Vert KWK \Vert_{2} \Vert K \Vert_{4} $$

  (2.31)

  because $K \in \mathfrak {S}_{4}$ [or any $\mathfrak {S}_{p}$, p>2, as a matter of fact]: just notice that by (2.16), (2.17)

  $$\begin{array}{@{}rcl@{}} \Vert K {\Vert_{4}^{4}}& = &{\sum}_{j = 0}^{\infty} \frac{1}{\vert z - z_{j} \vert^{2}} \leq \\ &\leq& 3/4 + 2 {\sum}_{k = 2}^{\infty} \left( \frac{2}{3} \right)^{2} \cdot \frac{1}{k^{2}} < 20 < \left( \frac{11}{5} \right)^{4}, \end{array} $$

  (2.32)

  so

  $$ \Vert K \Vert_{4} \leq \frac{11}{5}; \quad \Vert K {\Vert_{4}^{2}} \leq 5. $$

  (2.33)

  Therefore we can claim the following.

Proposition 2.3

Under the assumptions ( 2.1 ), ( 2.2 ), $0 < \alpha < \frac {1}{2}$ , suppose that N _∗ =N _∗ (α) is chosen in such a way that

$$ M(\alpha) \frac{\log en}{n^{2\alpha}} \leq \frac{1}{2} \quad \text{ for all} \quad n \geq N_{*}. $$

(2.34)

Then for n>N _∗ (α) if $z \in \partial \mathcal {D}_{n}$ all the operators U _j ∈( 2.8 ) are of the trace class, their norms satisfy inequalities

$$\begin{array}{@{}rcl@{}} \Vert U_{j} \Vert_{1} &\leq& 2 \left[ M(\alpha) \frac{\log en}{n^{2\alpha}}\right]^{j}, \quad j \geq 2, \end{array} $$

(2.35)

$$\begin{array}{@{}rcl@{}} \Vert U_{1} \Vert_{1} & =& \Vert R^{0} W R^{0} \Vert_{1} \leq \frac{5 M(\alpha) \log en}{n^{2\alpha}} \end{array} $$

(2.36)

and the Neumann - Riesz series for the difference of two resolvents

$$\begin{array}{@{}rcl@{}} R - R^{0} = {\sum}_{j = 1}^{\infty} U_{j} \end{array} $$

(2.37)

converges by the $\mathfrak {S}_{1}$ -norm and

$$\begin{array}{@{}rcl@{}} \Vert R - R^{0} \Vert_{1} \leq 7 M(\alpha) \frac{\log en}{n^{2\alpha}}\, \end{array} $$

(2.38)

and

$$\begin{array}{@{}rcl@{}} \left\Vert {\sum}_{j = m}^{\infty} U_{j} \right\Vert_{1} \leq 4 \left( M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^{m},\quad m \geq 2. \end{array} $$

(2.39)

Proof

Inequality (2.35) is identical with (proven) line (2.30). (2.36) come if we combine (2.32), (2.31), and (2.14). Therefore, for m≥2, by (2.35) and (2.34),

$$ \left\Vert {\sum}_{j = m}^{\infty} U_{j} \right\Vert_{1} \leq 2 {\sum}_{j = m}^{\infty} \left( M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^{j} \leq 4 \left( M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^{m}. $$

(2.40)

Then by (2.36)

$$\begin{array}{@{}rcl@{}} &&\Vert R - R^{0} \Vert_{1} \leq \Vert U_{1} \Vert_{1} + \left\Vert {\sum}_{j = 2}^{\infty} U_{j} \right\Vert_{1} \leq \\ &&\leq M(\alpha) \frac{\log en}{n^{2\alpha}} \cdot \left(5 + 4 M(\alpha) \frac{\log en}{n^{2\alpha}} \right) \leq 7M(\alpha) \frac{\log en}{n^{2\alpha}}. \end{array} $$

(2.41)

□

3 Deviations of Eigenvalues of The Harmonic Oscillator Operator and its Perturbations

• 1 Although the constructions and methods of this section are general and applicable to many operators with discrete spectrum and their perturbations, we will focus later in this section on the case of Harmonic Oscillator operator (2.1) and its functional representation

  $$ L^{0} y = - y^{\prime \prime} + x^{2} y $$

  (3.1)

  in $L^{2}(\mathbb {R})$.

  The Riesz-Neumann Series (2.37), (2.8) and (2.9) — as soon as its convergence in $\mathfrak {S}_{1}$ is properly justified — can be used to evaluate eigenvalues of the operator L=L ⁰+W.

  Under proper conditions, if n≥N _∗, the operator L has the only eigenvalue λ _n in H _n ; moreover, λ _n is simple and $\lambda _{n} \in \mathcal {D}_{n}$. Therefore, both of the projections

  $$\begin{array}{@{}rcl@{}} {P_{n}^{0}}=\frac{1}{2\pi i} \int\limits_{\partial \mathcal{D}_{n}} R^{0}(z) \, dz = \left\langle \cdot, h_{n} \right\rangle h_{n} \end{array} $$

  (3.2)

  and

  $$\begin{array}{@{}rcl@{}} P_{n} = \frac{1}{2\pi i} \int\limits_{\partial \mathcal{D}_{n}} R(z) \, dz = \left\langle \cdot, \psi_{n} \right\rangle \phi_{n} \end{array} $$

  (3.3)

  are of rank 1. [In (3.3), ϕ _n is an eigenfunction of L and ψ _n is an eigenfunction of L ^∗=L ⁰+W ^∗, with an eigenvalue $\mu _{n} = \overline {\lambda }_{n}$ in $\mathcal {D}_{n}$. We will not use this specific information so nothing more is explained now.]

  Therefore,

  $$\begin{array}{@{}rcl@{}} &&\text{Trace} {P_{n}^{0}} = \text{Trace} \, P_{n} = 1, \end{array} $$

  (3.4)
  $$\begin{array}{@{}rcl@{}} &&\text{Trace} \frac{1}{2\pi i} \int\limits_{\partial \mathcal{D}_{n}} (R(z) - R^{0}(z)) \, dz = 0. \end{array} $$

  (3.5)

  and

  $$\begin{array}{@{}rcl@{}} z_{n} & =& \text{Trace} \, \frac{1}{2\pi i} \int\limits_{\partial \mathcal{D}_{n}} z R^{0}(z) \, dz = 2n + 1, \end{array} $$

  (3.6)
  $$\begin{array}{@{}rcl@{}} \lambda_{n} & =& \text{Trace} \, \frac{1}{2\pi i} \int\limits_{\partial \mathcal{D}_{n}} z R(z) \, dz, \end{array} $$

  (3.7)

  So (3.4) to (3.7) imply

  $$ \lambda_{n} - z_{n} = \text{Trace} \, \frac{1}{2\pi i} \int\limits_{\partial \mathcal{D}_{n}} (z - z_{n}) [R(z) - R^{0}(z)] \, dz ={\sum}_{j = 1}^{\infty} T_{j}(n) $$

  (3.8)

  where we put [with $z_{n} = {\lambda _{n}^{0}} = 2n + 1$]

  $$ T_{j}(n) = T_{j}(n; W) = \frac{1}{2\pi i} \text{Trace} \, \int\limits_{\partial \mathcal{D}_{n}} (z - z_{n}) U_{j}(z) \, dz $$

  (3.9)

  Proposition 2.3 is used in (3.8), (3.9). Trace is a linear bounded functional of norm 1, on the space $\mathfrak {S}_{1}$ of trace-class operators ([15, 22]) . It implies the following.

Corollary 3.1

Under the assumptions of Proposition 2.3, with n≥N _∗ , we have

$$ \vert T_{j}(n) \vert \leq \left[ M(\alpha) \frac{\log en}{n^{2\alpha}}\right]^{j}, \quad j \geq 2 $$

(3.10)

and

$$ \vert T_{1}(n) \vert \leq \frac{9}{4} M(\alpha) \frac{\log en}{n^{2\alpha}} $$

(3.11)

Proof

With |Trace A|≤∥A∥₁ and

$$\begin{array}{@{}rcl@{}} &&\vert z - z_{n} \vert \leq \frac{1}{\sqrt{2}}, z \in \partial \mathcal{D}_{n},\\ &&\text{length}(\mathcal{D}_{n}) = 4 \end{array} $$

(3.12)

rough estimates of integrals (3.9) with j≥2 and j=1 based on (2.35) and (2.36) lead to (3.10) and (3.11). □

Corollary 3.2

Under the assumptions of Proposition 2.3, with n≥N _∗ ,

$$ \lambda_{n} = (2n + 1) + {\sum}_{j = 1}^{q} T_{j}(n) + r_{q}(n), \quad q \geq 1, $$

(3.13)

where

$$ \vert r_{q}(n) \vert \leq 2 \left( M(\alpha) \frac{\log en}{n^{2\alpha}} \right)^{q + 1} $$

(3.14)

Proof

The presentation of λ _n and the inequality follow from (3.8) and (2.39) if we put m=q+1 in (2.39) and notice that $2 \sqrt {2} < \pi$ when we multiply the constant factors in inequalities. □

• 2 Analysis of the function N _∗(α). This function is determined by the inequality (2.34). Later we consider potentials with the coupling coefficient s [see (4.2), (4.3)] so it is useful to know the behavior of X=X _β (t), the solution of the equation

  $$ t \frac{\log eX}{X^{\beta}} = \frac{1}{2}, \quad \beta = 2\alpha, \text{for large} t. $$

  (3.15)

  Let us rewrite it as

  $$\begin{array}{@{}rcl@{}} \tau \log Y = Y, \text{where}\, Y &= &(eX)^{\beta} \end{array} $$

  (3.16)
  $$\begin{array}{@{}rcl@{}} \text{and} \,\tau &= &\frac{2}{\beta} e^{\beta}t \end{array} $$

  (3.17)

  The (3.16) has two solutions

  $$ y(\tau) = 1 + \frac{1}{\tau} + O \left( \frac{1}{\tau^{2}} \right), \qquad\qquad\tau \to \infty $$

  (3.18)

  and

  $$ Y(\tau) \to \infty, \qquad\qquad\qquad\qquad\qquad\tau \to \infty. $$

  (3.19)

Lemma 3.3

The solution Y∈( 3.19 ) has an asymptotic

$$ Y(\tau) = \tau \log \tau \cdot (1 + r(\tau)) $$

(3.20)

where

$$ r(\tau)= \frac{\log \log \tau}{\log \tau} \left( 1 + o(1) \right) $$

(3.21)

so for any δ we can find τ ^∗ such that

$$ Y(\tau) \leq \tau \log \tau + \tau(1 + \delta) \log \log \tau, \quad \tau \geq \tau^{*}. $$

(3.22)

or τ _∗ <τ ^∗ such that

$$ Y(\tau) \leq (1 + \delta) \tau \log \tau, \quad \tau \geq \tau_{*}. $$

(3.23)

Proof

If we look for r≥0, in (3.20), which solves (3.16) we have:

$$ \tau \log \tau [1 + r] = \tau[\log \tau + \log \log \tau + \log(1 + r)] $$

(3.24)

or

$$ r = \varphi(r), \quad \varphi(X) = \xi + \eta \log (1 + X), \quad r > 0 $$

(3.25)

where

$$ \xi= \frac{\log \log \tau}{\log \tau}, \quad \eta = \frac{1}{\log \tau} $$

(3.26)

For any $\displaystyle 0 < \delta \leq \frac {1}{2}$ we can choose τ ^∗ such that

$$ 0 < \xi \leq \frac{\delta}{2}, \quad 0 < \eta < \frac{\delta}{2} \quad \text{ if} \tau \geq \tau^{*}. $$

(3.27)

Then the function φ, φ:[0,δ]→[0,δ] is a contraction mapping, and (3.25) has the unique solution

$$ r = r(\tau), \quad 0 < r(\tau) \leq \delta. \quad $$

(3.28)

Therefore,

$$ r = \frac{\log \log \tau}{\log \tau} + \frac{\rho}{\log \tau},\quad 0 < \rho \leq \delta. $$

(3.29)

This implies (3.21) with

$$ \frac{\rho}{\log \log \tau} = o(1). $$

(3.30)

□

Corollary 3.4

The solution X(t) of ( 3.15 ), 0<β≤1, goes to $\infty$ when $t \to \infty$ and

$$ X(t) \leq \left( 2t \log \frac{At}{\beta} \right)^{1/\beta}, \quad A \text{ an absolute constant,} $$

(3.31)

if t is large enough.

Proof

If we put (3.17) into (3.22) or (3.23) elementary simplifications give the inequality (3.31). □

• 3 Inequalities (2.18) and (2.30) guarantee that we can use the representation (2.8) and eventually “asymptotics” (3.13) if

  $$ C_{0} M(\alpha) \frac{\log e n}{n^{2\alpha}} \leq \frac{1}{2}, \quad C_{0} \in \text{2.2} $$

  (3.32)

  and M(α) by (2.25) is chosen as

  $$ M(\alpha) = \left[ 6 + \frac{4/3}{1- 2 \alpha} + \frac{1}{3\alpha} \right]. $$

  (3.33)

  Then (3.31), with β=2α<1, t=2C ₀ M(α), implies that N _∗ can be chosen as

  $$ N_{*} = N_{*}(C_{0}; \alpha) = \left[ 2 C_{0} M(\alpha) \log \left(\frac{A}{2\alpha} 2 C_{0} M(\alpha) \right) \right]^{1/(2\alpha)} $$

  (3.34)

  Now if α is fixed we are interested in the dependence of N _∗ on C ₀∈(2.2).

• 4 Recall that if W is a multiplier-operator

  $$ Wf = w(x) f(x), \text{ with} w \in L^{p}(\mathbb{R}^{1}), \quad 1 \leq p < \infty, $$

  (3.35)

  then as we observed and used in [19]

  $$ w_{jk} = \left\langle W h_{j}, h_{k} \right\rangle = {\int}_{-\infty}^{\infty} w(x) h_{j}(x) h_{k}(x) \, dx $$

  (3.36)

  so by Hölder inequality

  $$ \vert w_{jk} \vert \leq \left\Vert w \right\Vert_{p} \cdot \left\Vert h_{j} \right\Vert_{2q} \left\Vert h_{k} \right\Vert_{2q}, \quad \frac{1}{p} + \frac{1}{q} = 1. $$

  (3.37)

  with

  $$ q > 1, \quad 2q > 2. $$

  (3.38)

  But

  $$ \left\vert h_{k}(x) \right\vert \leq C k^{-1/12} $$

  (3.39)

  so

  $$\begin{array}{@{}rcl@{}} \int \vert h_{k} (x) \vert^{2q} \, dx &=& \int \vert h_{k} (x) \vert^{2} \vert h_{k} (x) \vert^{2(q-1)} \, dx2\\ & \leq& C^{2(q-1)} k^{-(q-1)/6} \int \vert h_{k}(x) \vert^{2} \, dx \end{array} $$

  (3.40)

  and

  $$ \left\Vert h_{k} \right\Vert_{2q} \leq C^{1/p} k^{-1/(12p)}, \quad p \geq 1. $$

  (3.41)

  This means that the matrix W satisfies the condition (2.2) with $\alpha = \frac {1}{12p}$ This observation was crucial in [19]; it gives a broad class of operators covered by (2.2) so our claims of this sections are applicable to the operators (3.35).

  But there are much better estimates of L ^p norms of the Hermite functions than (3.41).

Lemma 3.5

As $n \to \infty$ ,

$$\begin{array}{@{}rcl@{}} \left\Vert h_{n} \right\Vert_{r} &\sim& n^{-\frac{1}{2} \left( \frac{1}{2} - \frac{1}{r} \right)}, \quad 1 \leq r < 4 \end{array} $$

(3.42a)

$$\begin{array}{@{}rcl@{}} \left\Vert h_{n} \right\Vert_{r} &\sim& n^{- \frac{1}{8}} \log n, \quad r = 4 \end{array} $$

(3.42b)

$$\begin{array}{@{}rcl@{}} \left\Vert h_{n} \right\Vert_{r} &\sim& n^{-\frac{1}{6} \left( \frac{1}{r} + \frac{1}{2} \right)}, \quad r > 4 \end{array} $$

(3.42c)

See [23, Lemma 1.5.2] for the sketch of the proof and further explanations of these claims.

Therefore, (3.41) could be improved. If p>2 then by (3.37) q<2, 2q<4 so

$$\begin{array}{@{}rcl@{}} \left\Vert h_{k} \right\Vert_{2q} \sim k^{-\frac{1}{2} \left( \frac{1}{2} -\frac{1}{2q} \right)} = k^{-\frac{1}{4p}},&& p > 2. \end{array} $$

(3.43)

For p=2 we have 2q=4 and

$$\begin{array}{@{}rcl@{}} \left\Vert h_{k} \right\Vert_{4} \sim k^{-\frac{1}{8}} \log k, && p = 2. \end{array} $$

(3.44)

Finally, if 1≤p<2 then 2q>4 so

$$\begin{array}{@{}rcl@{}} \left\Vert h_{k} \right\Vert_{2q} \sim k^{-\frac{1}{6} \left( \frac{1}{2q} + \frac{1}{2} \right)} = k^{-\frac{1}{12} \left(2 - \frac{1}{p} \right)}, &&1 \leq p <2 \end{array} $$

(3.45)

All these estimates are used in Theorem 4.1, — see Section 4 below.

Of course, (3.39) shows that δ-potentials

$$ w(x) = {\sum}_{k = 1}^{m} c_{k} \delta (x - b_{k}), \quad m \leq \infty, \quad M \equiv {\sum}_{k = 1}^{m} \vert c_{k} \vert < \infty , $$

(3.46)

are good for us as well; in this case,

$$ \left\langle W h_{j}, h_{i} \right\rangle = {\sum}_{k = 1}^{m} c_{k} h_{j}(b_{k}) h_{i}(b_{k}), \quad \left\vert W_{ji} \right\vert \leq CM j^{-1/12} i^{-1/12}, \quad i, j \geq 1, $$

(3.47)

and we can give a trace-class version of Lemma 2.1.

Remark 3.6

Under the conditions (3.46), (2.10), if $z \in H_{n} \setminus \mathcal {D}_{n}$ then KWK is a trace class operator, and

$$ \Vert KWK \Vert_{1} \leq C_{12} \frac{M}{n^{1/6}} $$

(3.48)

where C ₁₂ is an absolute constant.

Proof

The proof follows if we observe that KWK is a sum of rank-one operators 〈⋅,g _k 〉g _k where

$$ g_{k} = \left( h_{j}(b_{k}) \frac{1}{\sqrt{z - z_{j}}} \right)_{j = 0}^{\infty} , \quad 1 \leq k \leq m $$

(3.49)

□

With more information on asymptotics of Hermite polynomials and Hermite functions we can be accurate in analysis of point-interaction potentials (3.46) and spectra of operators L ⁰+W, L ⁰∈(3.1), or — equivalently — (2.1). This is the main goal of this paper and its forthcoming extension. Now we go to detailed analysis of these operators.

4 Two-point Interaction Potentials

• 1 Now we apply general constructions of Sections 2, 3 to the case of the two-point interaction potentials

  $$ w(x) = c^{+} \delta \left( x - b \right) + c^{-} \delta \left( x + b \right), \quad \quad b > 0 $$

  (4.1)

  and particular cases of an odd potential

  $$\begin{array}{@{}rcl@{}} s v^{o}(x), \quad s \in \mathbb{C} \quad \text{ where}\quad\quad v^{o}(x) = \delta \left( x - b \right) - \delta \left( x + b \right) \end{array} $$

  (4.2)

  and an even potential

  $$\begin{array}{@{}rcl@{}} t v^{e}(x), \quad t \in \mathbb{C} \quad \quad v^{e}(x) = \delta \left( x - b \right) + \delta \left( x + b \right) \end{array} $$

  (4.3)

  — see [5, 14].

  Of course, for any odd potential (3.35) or (3.46), not just for v ⁰∈(4.2), the matrix elements w _{j k} have the property (4.6). Indeed, with parity

  $$\begin{array}{@{}rcl@{}} w_{jk} &=& \left\langle w(x) h_{j}(x),\, h_{k}(x) \right\rangle = \end{array} $$

  (4.4)
  $$\begin{array}{@{}rcl@{}} &=& \left\langle w(-x) h_{j}(-x),\, h_{k}(-x) \right\rangle = -(-1)^{j + k} w_{jk} \end{array} $$

  (4.5)

  so

  $$ w_{jk} = 0 \text{ if} j + k \text{ even.} $$

  (4.6)

  If, however, w in (3.35) or (3.46) is even then we conclude

  $$ w_{jk} = 0 \text{ if} j + k \text{ odd.} $$

  (4.7)

  These observations lead to information on complex eigenvalues of L=L ⁰+W.

Theorem 4.1

Let the potential

$$\begin{array}{@{}rcl@{}} w(x) \in L^{p}, \, \, 1 \leq p < \infty, \quad \nu& =& \left\Vert w \right\Vert_{p}, \text{ or} \end{array} $$

(4.8)

$$\begin{array}{@{}rcl@{}} w(x) = {\sum}_{k = 1}^{\infty} c_{k} \delta \left( x - b_{k} \right), \quad \nu &=& \sum \left\vert c_{k} \right\vert < \infty, \end{array} $$

(4.9)

be $\mathcal {P}\mathcal {T}$ , i.e., $w(-x) = \overline {w(x)}$ . Then the operator

$$ L = L^{0} + W = - \frac{d^{2}}{dx^{2}} + x^{2} + w $$

(4.10)

has at most finitely many non-real eigenvalues, if any, and their number does not exceed N ^∗ , where

$$\begin{array}{@{}rcl@{}} N^{*} &=& D\left (\nu \log (1 + \nu) \right)^{2p}, \qquad \qquad p > 2 \end{array} $$

(4.11a)

$$\begin{array}{@{}rcl@{}} N^{*} &=& D^{*} \left (\nu \log^{2} (1 + \nu) \right)^{4}, \qquad\quad p = 2 \end{array} $$

(4.11b)

$$\begin{array}{@{}rcl@{}} N^{*} &=& D \left (\nu \log (1 + \nu) \right)^{\frac{3}{\left(1 - \frac{1}{2p} \right)}}, \qquad 1 \leq p < 2 \end{array} $$

(4.11c)

$$\begin{array}{@{}rcl@{}} N^{*} &=& D_{*}\left(\nu \log (1 + \nu) \right)^{6},\qquad\qquad \text{in the case (4.9)} \end{array} $$

(4.11d)

D ^∗ , D _∗ are absolute constants, and D=D(p) does not depend on the norm ν.

Proof

By the estimates in Corollary 3.1 and in (3.34) we can use the series (3.8) to evaluate λ _n if n≥N ^∗, i. e., all eigenvalues in a half-plane z:R e z≥2N ^∗ – see (2.4), (2.5). The lemmas which follows explain that under the assumptions of the theorem every term T _j (3.8), (3.9) is real. But total number of all other eigenvalues in (z:R e z<2N ^∗} i s N ^∗ as Proposition 2 in [19] explains. So the number of non-real eigenvalues if any does not exceed N ^∗. Corollary 3.1 and Lemma 3.3 guarantee that N ^∗ could be chosen as (4.11. a – d) indicate because Lemma 3.5 leads good estimates of matrix elements (3.36). □

Lemma 4.2

Let w∈( 4.10 ) be a $\mathcal {P}\mathcal {T}$ -potential, i.e.,

$$ w (-x) = \overline{w(x)}, \quad x \in \mathbb{R}^{1}. $$

(4.12)

Then for n≥N ^∗ ∈( 4.11 ) all

$$ T_{j}(n; W), \quad 1 \leq j, \, \, \text{ are real} $$

(4.13)

Proof

If w=p+i q∈(4.12), p,q real, then p is even and q is odd so by (4.4) – (4.7),

$$w(k, \ell) = \left\{\begin{array}{ll}p(k, \ell), & \text{if} k + \ell \text{ even}\\ i q (k, \ell), & \text{if} k + \ell \text{ odd.} \end{array}\right. $$

By (3.9)

$$ T_{j}(n) = T_{j}(n; W) = \frac{1}{2\pi i} \,\text{Trace}\, \int\limits_{\partial \mathcal{D}_{n}} \, \, (z - z_{n}) U_{j}(z) \, dz $$

(4.14)

where

$$ U_{j} = R^{0} W R^{0} W \dotsb W R^{0} $$

(4.15)

with j “letters” W and j+1 “letters” R ⁰ in this “word” U. All these operators are of trace class [see Corollary 3.1] so Trace U _j is a sum of integrals of the diagonal elements (U _j ) _{m m} which in turn are sums of matrix elements $\displaystyle {\sum }_{g} u(g, z)$, where $g = \left \lbrace g_{1}, g_{2}, \dotsc , g_{j-1} \right \rbrace \in \mathbb {Z}_{+}^{j-1}$ and

$$ u(g, z) \equiv \frac{1}{z - z_{m}} \cdot W(m, g_{1}) \cdot \frac{1}{z - z_{g_{1}}} \cdot W(g_{1}, g_{2}) \cdot \frac{1}{z - z_{g_{2}}} \dotsb W(g_{j-1}, m) \cdot \frac{1}{z - z_{m}} $$

(4.16)

If we put g ₀=g _j =m we have

$$ {\sum}_{t = 0}^{j-1} (g_{t + 1} - g_{t}) = g_{j} - g_{0} = 0 $$

(4.17)

The sum of all these differences in (4.17) is even (zero): so

$$ \gamma^{-} \equiv \# {\Gamma}^{-}, \quad {\Gamma}^{-} \equiv\left\lbrace t: g_{t + 1} - g_{t} \text{ odd} \right\rbrace $$

(4.18)

should be even. Put

$$ \gamma^{+} \equiv \# {\Gamma}^{+}, \quad {\Gamma}^{+} \equiv \left\lbrace t: g_{t + 1} - g_{t} \text{ even} \right\rbrace, $$

(4.19)

and

$$ p(g) \equiv {\prod}_{t = 0}^{j-1} iW(g_{t}, g_{t + 1}). = (-1)^{q} \cdot \text{(real number)} $$

(4.20)

Then

$$ p(g) = \left( {\prod}_{t \in {\Gamma}^{-}} W(g_{t}, g_{t + 1}) \right) \left( {\prod}_{t \in {\Gamma}^{+}} W(g_{t}, g_{t + 1})\right) = i^{\gamma^{-}} \cdot (\text{real number}) $$

(4.21)

and by (4.18) p(g) is real.

By (4.16)

$$\begin{array}{@{}rcl@{}} u(g, z) = p(g) \cdot F_{g}(z), \text{ where} \end{array} $$

(4.22)

$$\begin{array}{@{}rcl@{}} F_{g}(z) = \frac{1}{(z - z_{m})^{2}} {\prod}_{t =1}^{j-1} \frac{1}{z - z_{g_{t}}} \end{array} $$

(4.23)

and this term brings into (4.14) the number-product p(g)J(g) where

$$ J(g) = \frac{1}{2\pi i} \int\limits_{\partial \mathcal{D}_{n}}(z - z_{n}) F_{g}(z) \, dz . $$

(4.24)

For any $g \in \mathbb {Z}_{+}^{(j - 1)}$ this integral J(g) is a real number [see the next lemma]. Therefore, T _j (n) — as a sum of (absolutely) convergent series with real terms — is a real number. □

This completes the proof of Proposition 4.1. We will need more specific information about integrals J(g)∈(4.24). The following is true.

Lemma 4.3

If m=n, and n≥N ^∗ (N ^∗ as defined in ( 4.11 )),

$$ J(g) = 0 \quad \text{ if at least one} \quad g (\tilde{t}) = n, $$

(4.25)

and

$$ J(g) = \left(\prod\limits_{t = 1}^{j-1}2 (n - g_{t}) \right)^{-1} \quad \text{otherwise} . $$

(4.26)

If m≠n,

$$ J(g) = 0 \,\text{if} \,\# \tau(g) \neq 2, \, \, \text{where} \tau(g) = \left\{ t: g_{t} = n \right\} $$

(4.27)

and

$$ J(g) = \frac{1}{4(n-m)^{2}}{\left(\prod\limits_{t \not\in \tau(g)}^{j-1}2 (n - g_{t}) \right)}^{-1} \text{if} \# \tau(g) = 2. $$

(4.28)

Proof

The integrand (4.23) of (4.24) could have a pole inside of $\mathcal {D}_{n}$ only at z _n =2n+1. In the cases (4.25) and (4.27) the pole’s order ≥2 or F _g (z) is analytic on $\overline {\mathcal {D}_{n}}$, so J(g)=0. In the cases (4.26), (4.28) the pole’s order is one and J(g) is the residue of F _g (z) at z _n . □

• 2 An odd potential v ^o in (4.2) As it is noticed in (4.6),

  $$\begin{array}{@{}rcl@{}} v^{0}_{jk} &=& \left\langle (\delta \left( x-b \right) - \delta \left( x + b \right)) h_{j}, h_{k} \right\rangle \\ & =& [1 - (-1)^{j + k}] a_{j} a_{k} \end{array} $$

  (4.29)
  $$\begin{array}{@{}rcl@{}} &=& \left\{\begin{array}{ll} 0, & \text{ if} j + k \text{ even} \\ 2 a_{j} a_{k}, & \text{ if} j + k \text{ odd} \end{array}\right. \end{array} $$

  (4.30)

  where

  $$\begin{array}{@{}rcl@{}} a_{k} = h_{k}(b), \quad k = 0, 1, \dotsc \end{array} $$

  (4.31)

  With b>0 fixed, from now on we will use (a _k ) as in (4.31). By Lemmas 4.2 and 4.3 for n≥N ^∗

  $$\begin{array}{@{}rcl@{}} T_{j}(n; v^{0}) \equiv T_{j}(n) = 0 \, \, \, \, \text{ if} j \text{ odd;} \end{array} $$

  (4.32)

  in particular

  $$ T_{1}(n) = 0 , \quad T_{3}(n) = 0. $$

  (4.33)

  To evaluate T ₂(n) we sum up (we did it in (4.16) in general setting) Cauchy integrals of functions

  $$ (z - z_{n}) \cdot \frac{1}{z - z_{m}} \cdot v_{mk}^{0} \cdot \frac{1}{z - z_{k}} \cdot v_{km}^{0} \cdot \frac{1}{z - z_{m}} $$

  (4.34)

  If m≠n it is analytic for any k so Cauchy integral is zero. If m=n

  $$ v_{mk}^{0} = 0 \quad \text{ if} n - k \text{ is even.} $$

  (4.35)

  Therefore, by Lemma 4.3, j=2, with z _n −z _k =2(n−k),

  $$\begin{array}{@{}rcl@{}} T_{2}(n; v^{0}) \equiv T_{2}(n) = \sum\limits_{\begin{array}{c} k = 0 \\ k - n \text{ odd}\end{array}}^{\infty} \frac{v_{nk}^{0} v_{kn}^{0}}{z_{n} - z_{k}} = \sum\limits_{\begin{array}{c} k = 0 \\ k - n \text{odd}\end{array}}^{\infty} \frac{2 a_{n} a_{k} \cdot 2 a_{k} a_{n}} {2(n - k)} = 2 {a^{2}_{n}} \widetilde{\sigma}(n)\\ \end{array} $$

  (4.36)

  where

  $$\begin{array}{@{}rcl@{}} \widetilde{\sigma}(n) = \sum\limits_{\begin{array}{c} k = 0 \\ n - k \text{odd}\end{array}}^{\infty} \frac{{a_{k}^{2}}} {n - k} \end{array} $$

  (4.37)

  The technical analysis of this sequence (4.37) and its variations is the core of our forthcoming paper which complements the present one. All the proofs will be given there. In meantime we can refer a reader to [18], Sections 5–8.

  It will bring us the proof of the main result of this paper:

Theorem 4.4

The operator

$$L = - \frac{d^{2}}{dx^{2}} + x^{2} + s [ \delta \left( x-b \right) - \delta \left( x + b \right)], b > 0, s \in \mathbb{C} $$

has a discrete spectrum σ(L).

There exists an absolute constant D such that with

$$ N^{*} = \left( D \vert s \vert \log e \vert s \vert \right)^{2} $$

(4.38)

all eigenvalues λ _n =λ _n (L) in the half-plane $\left \lbrace z \in \mathbb {C}: \Re z > N^{*} \right \rbrace$ are simple, and

$$ \lambda_{n} = (2n + 1) + s^{2} \, \frac{\kappa(n)}{n} + \widetilde{\rho}(n), \quad \vert \widetilde{\rho}(n) \vert \leq C\frac{\log n}{n^{3/2}} $$

(4.39)

where

$$ \kappa_{n} = \frac{1}{2\pi} \left[ (-1)^{n + 1} \sin (2 b \sqrt{2n}) - \frac{1}{2} \sin(4b \sqrt{2n}) \right] $$

(4.40)

The proof of the theorem is based on the following lemma.

Lemma 4.5

With $\widetilde {\sigma }(n) \in$ ( 4.37 )

$$\begin{array}{@{}rcl@{}} \widetilde{\sigma}(n) & =& (-1)^{n + 1}\frac{1}{2} \frac{\sin (2 b \sqrt{2n})}{\sqrt{2n}} + \rho(n), \end{array} $$

(4.41)

$$\begin{array}{@{}rcl@{}} \vert \rho(n) \vert &\leq& C \frac{\log n}{n} \end{array} $$

(4.42)

An even potential v ^e∈(4.3) Recall (4.7); now

$$ v^{e}_{jk} = [1 + (-1)^{j + k}] a_{j} a_{k} $$

(4.43)

$$\begin{array}{@{}rcl@{}} =\left\{\begin{array}{ll} 0, \text{ if} j + k \text{odd}\\ 2 a_{j} a_{k}, \text{ if} j + k \text{ even} \end{array}\right. \end{array} $$

(4.44)

Therefore, by Lemma 4.3, j=1,

$$ T_{1}(n; v^{e}) \equiv T_{1}(n) = v_{nn}^{e} = 2 {a_{n}^{2}} $$

(4.45)

and [compare (4.32) to (4.46)]

$$\begin{array}{@{}rcl@{}} T_{2}(n; v^{e}) \equiv T_{2}(n) &=& 2 {a_{n}^{2}} \sigma^{\prime}(n), \end{array} $$

(4.46)

$$\begin{array}{@{}rcl@{}} \sigma^{\prime}(n) & =& \sideset{~}{\prime}{\sum\limits}_{\begin{array}{c}k = 0\\ n - k \text{even}\end{array}}^{\infty} \frac{{a_{k}^{2}}}{n - k} , \end{array} $$

(4.47)

where $\sum \nolimits ^{\prime }$ means that k≠n.

But for the even potential there is no trivial claim T ₃(n)=0. We could make formal references to Lemma 4.3 but let us again look into those terms which form the sum-trace T ₃(n). We integrate functions

$$ F = (z - z_{n}) \cdot \frac{1}{z - z_{m}} \cdot 2a_{m} a_{k} \cdot \frac{1}{z - z_{k}} \cdot 2 a_{k} a_{\ell} \cdot \frac{1}{z - z_{\ell}} \cdot 2 a_{\ell} a_{m} \cdot \frac{1}{z - z_{m}} $$

(4.48)

excluding (by (4.44)) triples (m,k,ℓ) if at least one of the differences m−k, k−ℓ, ℓ−m is odd.

If m=n then we can take only k,ℓ≠n, otherwise the order of the pole at z _n would be ≥2 and Cauchy integral (4.24) be zero. Then the partial sum of (4.14) over triples

$$\left\lbrace (m, k, \ell) \vert m = n, k \neq n, \ell \neq n, k - n, \ell - n \text{ even} \right\rbrace $$

would be

$$\begin{array}{@{}rcl@{}} 2 {a_{n}^{2}} {\sum\limits}_{\begin{array}{c}k, \ell \\ n - k, \\ n - \ell \text{even}\end{array}}^{\prime} \frac{{a_{k}^{2}} a_{\ell}^{2}}{(n - k)(n - \ell)} = \end{array} $$

(4.49)

$$\begin{array}{@{}rcl@{}} = 2 {a_{n}^{2}} \left( \sigma^{\prime}(n) \right)^{2}. \end{array} $$

(4.50)

If m≠n Cauchy integral of F∈(4.48) is not zero only if k=ℓ=n, i.e., if we have two (and only two) zeros in the denominator to balance the factor (z−z _n ). This set of triples

$$ \left\{ (m, k, \ell) \vert m \neq n, k= \ell = n, m - n \text{ even} \right\} $$

(4.51)

leads to the subsum in T ₃(n) coming from (4.48)

$$ 2 {a_{n}^{4}} \sideset{~}{\prime}{\sum\limits}_{\begin{array}{c}m = 0 \\ m - n \text{even}\end{array}}^{\infty} \frac{{a_{m}^{2}}}{(n - m)^{2}} = 2 {a_{n}^{4}} \tau^{\prime}(n) $$

(4.52)

If we combine (4.49) – (4.52) we conclude that

$$ T_{3}(n; v^{e}) = 2 {a_{n}^{2}} \left[ \sigma^{\prime}(n)^{2} + {a_{n}^{2}} \tau^{\prime}(n) \right] $$

(4.53)

We will analyze the sequences $\sigma ^{\prime }$, $\tau ^{\prime }$ in a forthcoming paper as well.