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.
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1 Introduction
The operator
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 \({\Lambda }^{0} = \{z_{n}\}_{n = 0}^{\infty }\),
and a compact resolvent
A normalized orthogonal system of eigenfunctions can be chosen as the Hermite functions
where
are the Hermite polynomials.
Spectral analysis of perturbed operators
with special W, in particular, the point interaction perturbations
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
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
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
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
Moreover, we essentially use the property of perturbations W∈(1.5) to have such a matrix
that for some α>0 there exists M>0 such that
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
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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.
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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.
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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
Proof
If \(z \in \partial \mathcal {D}_{n}\), i.e,
then
and if |n−j|≥2,
with
The sum of three terms for j=n,n±1 in (2.19) by (2.16) does not exceed
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
The first integral (after the change of variables x=n ξ) is
The second integral in (2.21) is equal to
If we collect the inequalities (2.20) to (2.23), we get (with 22α≤2)
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
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
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. □
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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
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
and the Neumann - Riesz series for the difference of two resolvents
converges by the \(\mathfrak {S}_{1}\) -norm and
and
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),
Then by (2.36)
□
3 Deviations of Eigenvalues of The Harmonic Oscillator Operator and its Perturbations
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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)$$ \lambda_{n} - z_{n} = \text{Trace} \, \frac{1}{2\pi i} \int\limits_{\partial \mathcal{D}_{n}} (z - z_{n}) [R(z) - R^{0}(z)] \, dz ={\sum}_{j = 1}^{\infty} T_{j}(n) $$(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
and
Proof
With |Trace A|≤∥A∥1 and
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 ∗ ,
where
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. □
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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
where
so for any δ we can find τ ∗ such that
or τ ∗ <τ ∗ such that
Proof
If we look for r≥0, in (3.20), which solves (3.16) we have:
or
where
For any \(\displaystyle 0 < \delta \leq \frac {1}{2}\) we can choose τ ∗ such that
Then the function φ, φ:[0,δ]→[0,δ] is a contraction mapping, and (3.25) has the unique solution
Therefore,
This implies (3.21) with
□
Corollary 3.4
The solution X(t) of ( 3.15 ), 0<β≤1, goes to \(\infty \) when \(t \to \infty \) and
if t is large enough.
Proof
If we put (3.17) into (3.22) or (3.23) elementary simplifications give the inequality (3.31). □
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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).
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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 \(n \to \infty \) ,
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
For p=2 we have 2q=4 and
Finally, if 1≤p<2 then 2q>4 so
All these estimates are used in Theorem 4.1, — see Section 4 below.
Of course, (3.39) shows that δ-potentials
are good for us as well; in this case,
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
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
□
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
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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)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
be \(\mathcal {P}\mathcal {T}\) , i.e., \(w(-x) = \overline {w(x)}\) . Then the operator
has at most finitely many non-real eigenvalues, if any, and their number does not exceed N ∗ , where
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.,
Then for n≥N ∗ ∈( 4.11 ) all
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),
By (3.9)
where
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
If we put g 0=g j =m we have
The sum of all these differences in (4.17) is even (zero): so
should be even. Put
and
Then
and by (4.18) p(g) is real.
By (4.16)
and this term brings into (4.14) the number-product p(g)J(g) where
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 )),
and
If m≠n,
and
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 . □
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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
has a discrete spectrum σ(L).
There exists an absolute constant D such that with
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
where
The proof of the theorem is based on the following lemma.
Lemma 4.5
With \(\widetilde {\sigma }(n) \in \) ( 4.37 )
An even potential v e∈(4.3) Recall (4.7); now
Therefore, by Lemma 4.3, j=1,
and [compare (4.32) to (4.46)]
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
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
would be
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
leads to the subsum in T 3(n) coming from (4.48)
If we combine (4.49) – (4.52) we conclude that
We will analyze the sequences \(\sigma ^{\prime }\), \(\tau ^{\prime }\) in a forthcoming paper as well.
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Acknowledgements
The author is indebted to Charles Baker and Petr Siegl for numerous discussions. Without their support this work would hardly be written, at least in a reasonable period of time. I am also thankful to Daniel Elton, Paul Nevai, Günter Wunner, and Miloslav Znojil for valuable comments and information related to topics of this manuscript.
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Mityagin, B.S. The Spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions. Int J Theor Phys 54, 4068–4085 (2015). https://doi.org/10.1007/s10773-014-2468-z
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DOI: https://doi.org/10.1007/s10773-014-2468-z