Non-real eigenvalues of the Harmonic Oscillator perturbed by an odd, two-point $\delta$-potential

TL;DR

This paper investigates how odd two-point delta potentials perturb the harmonic oscillator, revealing that non-real eigenvalues proliferate infinitely as the potential's strength increases.

Contribution

It provides a detailed analysis of the spectrum of the harmonic oscillator with odd two-point delta perturbations, highlighting the unbounded growth of non-real eigenvalues.

Findings

Number of non-real eigenvalues tends to infinity as potential strength increases

Derived a formula for eigenvalues of the perturbed operator

Showed the spectrum's behavior under odd delta interactions

Abstract

In this paper, we consider the perturbations of the Harmonic Oscillator Operator by an odd pair of point interactions: $z (\delta(x - b) - \delta(x + b))$. We study the spectrum by analyzing a convenient formula for the eigenvalue. We conclude that if $z = ir$, $r$ real, as $r \to \infty$, the number of non-real eigenvalues tends to infinity.

Figures

Equations

$$L_{\text{HO}}^{0}y(x)$$

$$D\left(L_{\text{HO}}^0\right)$$

$$\mathrm{Sp}(L_{\text{HO}}^0)=2{\mathbb{N}}_0+1,{\mathbb{N}}_0\equiv \mathbb{N}\cup \{0\}.$$

$$h_n(x)\equiv \frac{1}{\sqrt{2^{n}n!\sqrt{\pi }}}{e}^{-x^2/2}{H}_n(x),$$

$$H_n(x)\equiv (-1{)}^n{e}^{x^2}\frac{d^n}{{dx}^n}\left(e^{-x^2}\right).$$

$$L_{\text{HO}}\left(z,b\right)y(x)=L_{\text{HO}}^{0}y(x)+z(\delta \left(x-b\right)-\delta \left(x+b\right))y(x),b>0,z\in \mathbb{C}.$$

$$\begin{array}{rl}{\mathfrak{t}}_{z,b}^{\text{HO}}(f(x),g(x))& ={\left(f^{\prime }(x),g^{\prime }(x)\right)}_{L^2(\mathbb{R})}+{\left(xf(x),xg(x)\right)}_{L^2(\mathbb{R})}\\ & +z\left(f(b)\overline{g(b)}-f(-b)\overline{g(-b)}\right).\end{array}$$

$${\left(L_{\text{HO}}\left(z,b\right)f(x),g(x)\right)}_{L^2(\mathbb{R})}={\mathfrak{t}}_{z,b}^{\text{HO}}(f(x),g(x)).$$

$$\underset{r\to \infty }{\lim }{\mathcal{N}}_{\text{HO}}(r)=\infty .$$

$$\begin{array}{rl}M(\nu ;b)& =\frac{1}{z^2},\text{ where }\\ M(\nu ;b)& \equiv \sqrt{\frac{2}{\pi }}\Gamma (-\nu )D_{\nu }^2(b)y_{\text{even}}(\nu ;b)y_{\text{odd}}(\nu ;b).\end{array}$$

$$-\frac{d^{2}y}{{dx}^2}+\left(\frac{x^2}{4}-\frac{1}{2}\right)y(x)=\nu y(x),x\in \mathbb{C},\nu \in \mathbb{C}.$$

$$-\frac{d^{2}y}{{dx}^2}+x^{2}y(x)=\lambda y(x).$$

$$L_{\text{PC}}\left(z,b\right)y(x)=-y^{\prime \prime }(x)+\left(\frac{x^2}{4}-\frac{1}{2}\right)y(x)+z\left[\delta \left(x-b\right)-\delta \left(x+b\right)\right]y(x).$$

$$\begin{array}{rl}{\mathfrak{t}}_{z,b}\left(f,g\right)& \equiv {\left(f^{\prime }(x),g^{\prime }(x)\right)}_{L^2(\mathbb{R})}+\frac{1}{4}{\left(xf(x),xg(x)\right)}_{L^2(\mathbb{R})}-\frac{1}{2}{\left(f(x),g(x)\right)}_{L^2(\mathbb{R})}\\ & +z\left(f(b)\overline{g(b)}-zf(-b)\overline{g(-b)}\right)\end{array}$$

$${\left(L_{\text{PC}}\left(z,b\right)f(x),g(x)\right)}_{L^2(\mathbb{R})}={\mathfrak{t}}_{z,b}\left(f,g\right).$$

$$L_{\text{PC}}\left(z,b\right)=\frac{1}{2}{T}^{-1}\circ L_{\text{HO}}\left(z\sqrt{2},\frac{b}{\sqrt{2}}\right)\circ T-\frac{1}{2}I,$$

$$\mathrm{Sp}(L_{\text{PC}}\left(z,b\right))=\frac{\mathrm{Sp}{L}_{\text{HO}}\left(z\sqrt{2},\frac{b}{\sqrt{2}}\right)-1}{2}.$$

$$\begin{array}{rl}\frac{1}{\Gamma (\zeta )}& =e^{\gamma \zeta }\prod _{n=1}^{\infty }\left(1+\frac{\zeta }{n}\right)e^{\zeta /n}\\ \gamma & =\underset{n\to \infty }{\lim }\left[\left(1+\frac{1}{2}+\cdots +\frac{1}{n}\right)-\log (n+1)\right].\end{array}$$

$$\ln \Gamma (z)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln (2\pi )+O\left(\frac{1}{r}\right),$$

$$\ln \left(M_{1/\Gamma }(r)\right):=\ln \left(\underset{|z|=r}{\sup }\left|\frac{1}{\Gamma (z)}\right|\right)\sim r\log r,r\to \infty .$$

$$\Gamma (2w)=\frac{2^{2w-1}}{\sqrt{\pi }}\Gamma (w)\Gamma \left(w+\frac{1}{2}\right)$$

$${\int }_0^{\infty }\exp (-z{t}^{\mu })t^{\zeta -1}dt=\frac{1}{\mu }\Gamma \left(\frac{\zeta }{\mu }\right)\frac{1}{z^{\zeta /\mu }},$$

$$-\frac{d^{2}y}{{dx}^2}+\left(\frac{x^2}{4}-\frac{1}{2}\right)y(x)$$

$$-\frac{d^{2}y}{{dx}^2}+\left(\frac{1}{4}{x}^2+a\right)y(x)$$

$$a=-\nu -\frac{1}{2}$$

$$D_{\nu }(x)\sim x^{\nu }{e}^{-x^2/4},x\to +\infty .$$

$$D_{\nu }(0)=\frac{2^{\nu /2}\sqrt{\pi }}{\Gamma \left(-\frac{\nu }{2}+\frac{1}{2}\right)},{\frac{\partial }{\partial x}\left(D_{\nu }(x)\right)|}_{x=0}=-\frac{2^{(\nu +1)/2}\sqrt{\pi }}{\Gamma \left(-\frac{\nu }{2}\right)},$$

$$D_{\nu }(x)=\{\begin{array}{ll}\frac{e^{-x^2/4}}{\Gamma (-\nu )}{\int }_0^{\infty }{t}^{-\nu -1}{e}^{-t^2/2-xt}dt,& \mathrm{Re}\nu <0,\\ \sqrt{\frac{2}{\pi }}{e}^{x^2/4}{\int }_0^{\infty }{e}^{-t^2/2}\cos \left(\frac{\pi \nu }{2}-xt\right)t^{\nu }dt,& \mathrm{Re}\nu >-1.\end{array}$$

$$-\frac{d^{2}y}{{dx}^2}+\left(\frac{x^2}{4}-\frac{1}{2}\right)y(x)=\nu y(x),x\in \mathbb{C},\nu \in \mathbb{C},$$

$$\mathrm{Wr}[D_{\nu }(x),D_{\nu }(-x)]$$

$$\mathrm{Wr}[D_{\nu }(x),D_{-\nu -1}(\pm ix)]$$

$$y_{\text{even}}(\nu ;x)$$

Full text

1. Introduction

1.1. Perturbations of the Harmonic Oscillator

We consider the harmonic oscillator operator,

[TABLE]

($\displaystyle\mathcal{H}^{k}(\mathbb{R})=\mathcal{W}^{k,2}(\mathbb{R})$ is a Sobolev space, and we allow $\displaystyle y^{\prime\prime}$ to be a distributional derivative.) The operator is of compact resolvent, and the spectrum is the positive odd integers:

[TABLE]

(For an operator $\displaystyle T$, $\displaystyle\operatorname{Sp}(T)$ denotes its spectrum.) The eigenfunction with eigenvalue $\displaystyle 2n+1$, $\displaystyle n\in\mathbb{N}_{0}$, is the $\displaystyle n$th Hermite function,

[TABLE]

where

[TABLE]

is the $\displaystyle n$th Hermite Polynomial. We now consider a perturbation of the harmonic oscillator operator,

[TABLE]

The operator’s construction is discussed in [MS16] (following [Kat95, Chapter VI]), where it is noted that the operator has compact resolvent [MS16, p. 5]. In particular, the operator is defined by the quadratic-form method, for the quadratic form $\displaystyle\mathfrak{t}^{\textnormal{HO}}_{z,b}$ with domain $\displaystyle\operatorname{\mathfrak{D}}\left(\mathfrak{t}^{\textnormal{HO}}_{z,b}\right)=\{f(x)\in\mathcal{H}^{1}(\mathbb{R}):xf(x)\in L^{2}(\mathbb{R})\}$ by

[TABLE]

In particular, for $\displaystyle f\in\operatorname{\mathfrak{D}}\left(L_{\text{HO}}\left(z,b\right)\right)\subset\operatorname{\mathfrak{D}}\left(\mathfrak{t}^{\textnormal{HO}}_{z,b}\right)$ and $\displaystyle g\in\operatorname{\mathfrak{D}}\left(\mathfrak{t}^{\textnormal{HO}}_{z,b}\right)$,

[TABLE]

This perturbation was studied in [MS16], [HCW14], [Mit15], and [Mit16], and a similar operator was studied in [Dem05]. In particular, [HCW14] and [Dem05] gave numerical evidence that when $\displaystyle z=ir$, non-real eigenvalues exist for large enough $\displaystyle r$, and [Mit15] proved that for $\displaystyle z=ir$, the number of non-real eigenvalues was bounded above by $\displaystyle M(\left|r\right|\log(e\left|r\right|))^{2}$. But how does the number of non-real eigenvalues change as $\displaystyle r$ increases? We show — and this is a main result of our paper — that $\displaystyle\mathcal{N}_{\textnormal{HO}}(r)=\#\left(\operatorname{Sp}(L_{\text{HO}}\left(ir,b\right))\setminus\mathbb{R}\right)$ increases to $\displaystyle\infty$.

Theorem 1**.**

For any fixed $\displaystyle b>0$,

[TABLE]

Our approach is based on finding a function $\displaystyle M(\nu,b)$ (meromorphic in $\displaystyle\nu$, holomorphic in $\displaystyle b$) such that $\displaystyle\nu\in\mathbb{C}\setminus\mathbb{N}_{0}$ is an eigenvalue of (the transformation of) $\displaystyle L_{\text{HO}}\left(z,b\right)$ 4 for the derivations),

[TABLE]

The zeros of $\displaystyle M(\nu,b)$ in the parameter $\displaystyle\nu$, in particular of its factor $\displaystyle D_{\nu}(b)$, become important to find and analyze the asymptotics of the eigenvalues for $\displaystyle\left|z\right|$ large. We observe that there are infinitely many zeros of $\displaystyle\mathsf{D}(\nu)=D_{\nu}(b)$ (see Section 5), where we streamline some arguments with the work of F. W. Olver ([Olv59], [Olv61]), with tighter results on the growth rates of $\displaystyle\mathsf{D}(\nu)$ than necessary. Around each zero $\displaystyle\lambda$ of $\displaystyle\mathsf{D}(\nu)$, for large $\displaystyle\left|z\right|$, we find solutions of (1.6) in a small neighborhood of $\displaystyle\lambda$, non-real for $\displaystyle z$ imaginary (see Sections 6 and 7); this is an important step in the completion of the proof (see Section 8).

1.2. Change of Variables

We make a change of variables, since the Weber differential equation, written in the form

[TABLE]

has its general solution far more studied than the the notation above,

[TABLE]

We define (with $\displaystyle z\in\mathbb{C}$, $\displaystyle b>0$)

[TABLE]

The corresponding quadratic form, $\displaystyle\mathfrak{t}_{z,b}$, has the same domain, $\displaystyle\operatorname{\mathfrak{D}}\left(\mathfrak{t}_{z,b}\right)\equiv\{f(x)\in\mathcal{H}^{1}(\mathbb{R}):xf(x)\in L^{2}(\mathbb{R})\}$, and is defined by

[TABLE]

and again, for $\displaystyle f\in\operatorname{\mathfrak{D}}\left(L_{\text{PC}}\left(z,b\right)\right)\subset\operatorname{\mathfrak{D}}\left(\mathfrak{t}_{z,b}\right)$ and $\displaystyle g\in\operatorname{\mathfrak{D}}\left(\mathfrak{t}_{z,b}\right)$,

[TABLE]

One may check that if $\displaystyle Sx=x\sqrt{2}$ is the dilation on the real line, and $\displaystyle Tf=f\circ S$ is the corresponding operator on $\displaystyle L^{2}(\mathbb{R})$, we have that

[TABLE]

and

[TABLE]

Hence, defining $\displaystyle\mathcal{N}_{\textnormal{PC}}(r)=\#\left(\operatorname{Sp}(L_{\text{PC}}\left(ir,b\right))\setminus\mathbb{R}\right)$, (1.5) is equivalent to the claim that $\displaystyle\lim_{r\to\infty}\mathcal{N}_{\textnormal{PC}}(r)=\infty$.

2. Reciprocal Gamma Function

Following complex-analysis convention (e.g., [Lev64, p. 27]), we define the entire function

[TABLE]

and its multiplicative inverse is the usual Gamma function, a meromorphic function with poles at the nonpositive integers. In particular, for $\displaystyle n$ a nonnegative integer, $\displaystyle\frac{1}{\Gamma(-n)}=0$. The Stirling approximation for the gamma function yields the estimate [Lev64, Chap.1, Sec. 11, p. 27], with $\displaystyle\left|z\right|=r$, and outside of circles of fixed width about the points in $\displaystyle-\mathbb{N}_{0}$,

[TABLE]

therefore

[TABLE]

(In this text, $\displaystyle f(x)\sim g(x)$ as $\displaystyle x\to+\infty$ means $\displaystyle\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$.) Some properties of the Gamma function on the positive real line are as follows.

Fact 2.1**.**

On $\displaystyle(0,\infty)$, $\displaystyle\log\Gamma(x)$ is positive and convex, and $\displaystyle\Gamma(1)=\Gamma(2)=1$; hence, for $\displaystyle x$ real and positive, $\displaystyle\Gamma(x)$ has a unique minimum in $\displaystyle[1,2]$. In particular, $\displaystyle\Gamma(x)$ is increasing on $\displaystyle[2,\infty)$.

See , e.g., [Con78, Section 7.7, p. 179] and [Olv74, Section 2.2, Exercise 2.4, p. 39].

Fact 2.2**.**

For $\displaystyle 2w\not\in-\mathbb{N}_{0}$,

[TABLE]

See, e.g., [Olv74, Chapter 2, Section 1, (1.08), p. 35].

Fact 2.3**.**

If $\displaystyle\operatorname{Re}\zeta>0$, $\displaystyle\mu>0$, and $\displaystyle\operatorname{Re}z>0$,

[TABLE]

where the implicit logarithms use the principal branch of the logarithm.

See, e.g., [Olv74, Chapter 2, Section 1, Exer. 1.1., p. 38].

3. The Weber Differential Equation and its Solutions

3.1. Notation

The Weber differential equation can be written in either of the forms

[TABLE]

These notations are equivalent under the rule

[TABLE]

and (3.2) is assumed throughout the rest of the paper. We will use (\theparentequation.i), as it is the choice of coordinates used by the relevant references [Dem05] and [MOS66], and because of a clearer connection to the harmonic oscillator. We mention (\theparentequation.ii) because of the frequent use of this form in the literature (e.g., [Olv74, Section 6.6], [Tem19], and [Dea66]).

The solutions of (3.1) are called parabolic cylinder functions. We now discuss some particular solutions.

3.2. Solutions decaying as $\displaystyle x\to\infty$: $\displaystyle D_{\nu}(x)$ or $\displaystyle U(a,x)$

One solution of (\theparentequation.i), denoted $\displaystyle D_{\nu}(x)$, is a solution of (\theparentequation.i) that decays as $\displaystyle x\to+\infty$; more precisely (e.g., [Tem19, Section 12.9(i), (12.9.1)])

[TABLE]

$\displaystyle D_{\nu}(x)$ may also be characterized by the values,

[TABLE]

the connection being derivable from the integral formulas (e.g., [MOS66, Section 8.1.4, p. 328]),

[TABLE]

As the coefficients of the differential equation (\theparentequation.i) are jointly continuous in $\displaystyle x$ and $\displaystyle\nu$, and holomorphic in each variable separately, and since the initial conditions are homomorphic in $\displaystyle\nu$, a standard continuation-of-parameters result (e.g., [Olv74, Section 5.3, Thm. 3.2, p. 146]) ensures that for each $\displaystyle x$, $\displaystyle D_{\nu}(x)$ is holomorphic in $\displaystyle\nu$.

If the Weber parabolic cylinder equation is written in $\displaystyle a$-notation, i.e. (\theparentequation.ii), then the function named $\displaystyle D_{\nu}(x)$ in $\displaystyle\nu$-notation is denoted $\displaystyle U(a,x)$.

In the sequel, we will abuse language and call $\displaystyle D_{\nu}(x)=U(a,x)$ “the” parabolic cylinder function.

3.3. Transformations of the parabolic cylinder function

Certain transformations of the parabolic cylinder function still satisfy the Weber differential equation (\theparentequation.i) (see, e.g.,[MOS66, Section 8.1.1, p. 324, and Section 8.1.3, p. 327]).

Fact 3.1**.**

For the differential equation

[TABLE]

solutions include $\displaystyle D_{\nu}(x)$, $\displaystyle D_{\nu}(-x)$, $\displaystyle D_{-\nu-1}(ix)$, and $\displaystyle D_{-\nu-1}(-ix)$. Some Wronskians include:

[TABLE]

Remark 3.2*.*

Since (3.5) has no $\displaystyle\frac{dy}{dx}$ term, the Wronskians of any two of its solutions are constant functions.

3.4. The even and odd solutions

We now present standard even and odd solutions to (\theparentequation.i), given by (e.g., [Olv74, Chap. 5, Exercise 3.5, pp. 147–148])

[TABLE]

For future reference, we note that from (\theparentequation.i) and (\theparentequation.ii), one sees that

[TABLE]

so these new solutions are also holomorphic in $\displaystyle x$ and $\displaystyle\nu$.

By (3.8) and (3.3), we may write $\displaystyle D_{\nu}(x)$ in terms of the even and odd solutions (e.g., [Tem19, Section 12.4]):

[TABLE]

3.5. (Non)interference of the zeros of different solutions

In the sequel, we fix $\displaystyle x$ and discuss the zeroes of $\displaystyle D_{\nu}(x)$ in the parameter $\displaystyle\nu$, and argue to what extent the zeroes in the parameter of $\displaystyle y_{\textnormal{even}}(\nu;x)$ and $\displaystyle y_{\textnormal{odd}}(\nu;x)$ do or do not interfere.

Lemma 3.3**.**

*Fix $\displaystyle x_{0}\in\mathbb{C}$.

1. (a).

If $\displaystyle\nu\not\in\mathbb{N}_{0}$, and $\displaystyle D_{\nu}(x_{0})=0$, then $\displaystyle y_{\textnormal{even}}(\nu;x_{0})\neq 0$ and $\displaystyle y_{\textnormal{odd}}(\nu;x_{0})\neq 0$. 2. (b).

if $\displaystyle n\in\mathbb{N}_{0}$, and $\displaystyle D_{n}(x_{0})=0$, then exactly one of $\displaystyle y_{\textnormal{even}}(n;x_{0})$, $\displaystyle y_{\textnormal{odd}}(n;x_{0})$ is $\displaystyle 0$.

Proof, Part (a).

Fix $\displaystyle\nu,x_{0}\in\mathbb{C}$, $\displaystyle\nu\not\in\mathbb{N}_{0}$, such that $\displaystyle D_{\nu}(x_{0})=0$. If $\displaystyle y_{\textnormal{even}}(\nu;x_{0})=0$, then by (3.9),

[TABLE]

Hence, the Wronskian of $\displaystyle y_{\textnormal{even}}(\nu;x)$ and $\displaystyle y_{\textnormal{odd}}(\nu;x)$ would be $\displaystyle 0$ at $\displaystyle x=x_{0}$. Yet by (3.8),

[TABLE]

and hence the Wronskian cannot vanish anywhere in $\displaystyle\mathbb{C}$; in particular, $\displaystyle y_{\textnormal{even}}(\nu;x_{0})$ and $\displaystyle y_{\textnormal{odd}}(\nu;x_{0})$ can never be simultaneously 0. Contradiction. Similarly if $\displaystyle y_{\textnormal{odd}}(\nu;x_{0})=0$. ∎

Proof, Part (b).

Note that if $\displaystyle n=2k$ is a nonnegative even integer, then by (3.9),

[TABLE]

so $\displaystyle D_{2k}(x)$ is a nonzero multiple of $\displaystyle y_{\textnormal{even}}(2k;x)$, and these functions have the same zeroes. The proof works similarly if $\displaystyle n=2k+1$ is a positive odd integer. ∎

4. Eigenvalue conditions

4.1. $\displaystyle L^{2}(\mathbb{R})$ solutions

We return to finding the eigenvalues of

[TABLE]

Fact 4.1** (Folklore).**

Fix $\displaystyle b>0$, $\displaystyle z\in\mathbb{C}$, and $\displaystyle y(x)\in\mathcal{H}^{1}(\mathbb{R})$. Then the conditions

1. (i)

$\displaystyle y(x)\in\operatorname{\mathfrak{D}}\left(L_{\text{PC}}\left(z,b\right)\right)$, and 2. (ii)

$\displaystyle y(x)$* is an eigenfunction of (4.1) with eigenvalue $\displaystyle\nu$, i.e.,*

[TABLE]

hold if and only if $\displaystyle y(x)$ is a $\displaystyle C^{\infty}$ solution of (3.5) on $\displaystyle(b,\infty)$, $\displaystyle(-b,b)$, and $\displaystyle(-\infty,-b)$, and we have the following jumps in the first derivative

[TABLE]

(For $\displaystyle c$ real, $\displaystyle f(x)$ a function on $\displaystyle\mathbb{R}$, $\displaystyle f(c\pm)=\lim_{x\to c^{\pm}}f(x)$.) Combining Fact 3.1 and Fact 4.1, we have the following.

Lemma 4.2**.**

Fix $\displaystyle b>0$ and $\displaystyle z\in\mathbb{C}$. If $\displaystyle y(x)$ is an eigenfunction of $\displaystyle L_{\text{PC}}\left(z,b\right)$ with eigenvalue $\displaystyle\nu$, then

[TABLE]

for some complex $\displaystyle\alpha,\beta,\sigma,\tau$.

Proof.

We choose a basis of solutions to (3.5) on each subinterval.

**On $\displaystyle[b,\infty)$: **

$\displaystyle\{D_{\nu}(x),D_{-\nu-1}(ix)\}$ is a basis because their Wronskian is nonzero (see (3.6b)).

**On $\displaystyle(-b,b)$: **

$\displaystyle\{y_{\textnormal{even}}(\nu;x),y_{\textnormal{odd}}(\nu;x)\}$ is a basis here. Note that by (3.8),

[TABLE]

and since $\displaystyle y_{\textnormal{even}}(\nu;x)$ and $\displaystyle y_{\textnormal{odd}}(\nu;x)$ satisfy the same differential equation, their Wronskian is therefore never $\displaystyle 0$, so we indeed have a basis.

**On $\displaystyle(-\infty,-b]$: **

We choose $\displaystyle\{D_{\nu}(-x),D_{-\nu-1}(ix)\}$ is a basis. The Wronskian is nonzero, because by the Chain Rule,

[TABLE]

so that again by (3.6b),

[TABLE]

Eigenfunctions are solutions to the unperturbed differential equation (3.5) on each subinterval, by Fact 4.1; therefore

[TABLE]

for some complex constants $\displaystyle\beta,t,\sigma,\tau,\alpha,s$. Yet we have the known asymptotic as $\displaystyle\left|x\right|\to\infty$, ([MOS66, Section 8.1.6, p. 331]),

[TABLE]

Applying this to $\displaystyle D_{-\nu-1}(\pm ix)$, if $\displaystyle\left|\arg x\right|<\frac{\pi}{4}$,

[TABLE]

Hence, in (4.4), line 3, $\displaystyle s=0$, otherwise the solutions would be growing in magnitude as $\displaystyle x\to+\infty$, which is incompatible with $\displaystyle y\in L^{2}(\mathbb{R})$. In the same way, we can show in (4.4), line 1, $\displaystyle t=0$, if we rewrite $\displaystyle D_{-\nu-1}(ix)=D_{-\nu-1}(-i\cdot(-x))$, so that as $\displaystyle x\to-\infty$, $\displaystyle\arg(-x)=0$ and we may apply the above asymptotic. ∎

4.2. Boundary conditions at $\displaystyle\pm b$

Suppose that $\displaystyle y(x)$ is indeed an eigenfunction of $\displaystyle L_{\text{PC}}\left(z,b\right)$ with eigenvalue $\displaystyle\nu$. Then by Lemma 4.2, we have that

[TABLE]

Yet functions in the domain of $\displaystyle L_{\text{PC}}\left(z,b\right)$ are continuous, so we must have

[TABLE]

and

[TABLE]

Similarly, the jump condition at $\displaystyle+b$ (i.e., (4.2) becomes

[TABLE]

The jump condition at $\displaystyle-b$ becomes (by function parity and the Chain Rule)

[TABLE]

Putting this all together, and letting

[TABLE]

we have that

[TABLE]

This has nontrivial solutions (i.e., the $\displaystyle\nu$-eigenspace is nontrivial) if and only if the determinant is nonzero, i.e., if and only if

[TABLE]

Yet recalling the definitions of $\displaystyle\mathcal{P}$, etc. (i.e., (4.10)), we see that

[TABLE]

By the decomposition of $\displaystyle D_{\nu}(x)$ into the even and odd terms, however, this becomes

[TABLE]

As the Wronskians of solutions to (\theparentequation.i) are constant functions (see Remark 3.2), zero can be chosen as the evaluation point, and we get

[TABLE]

Similarly, one has

[TABLE]

Altogether, then, (4.12) becomes

[TABLE]

By using the Gamma-function double-angle formula (e.g., Fact 2.2, (2.2)),

[TABLE]

and applying with $\displaystyle w=-\frac{\nu}{2}$, $\displaystyle\nu\not\in\mathbb{N}_{0}$, we have

[TABLE]

Yet (4.14) holds for $\displaystyle\nu\in\mathbb{N}_{0}$ as well. If $\displaystyle\nu=2k$ is nonnegative and even, then both $\displaystyle\frac{1}{\Gamma(-\nu)}=\frac{1}{\Gamma(-2k)}$ and $\displaystyle\frac{1}{\Gamma(-\frac{\nu}{2})}=\frac{1}{\Gamma(-k)}$ are $\displaystyle 0$, and if $\displaystyle\nu=2k+1$ is positive and odd, then both $\displaystyle\frac{1}{\Gamma(-\nu)}=\frac{1}{\Gamma(-2k-1)}$ and $\displaystyle\frac{1}{\Gamma\left(-\frac{\nu}{2}+\frac{1}{2}\right)}=\frac{1}{-k}$ are $\displaystyle 0$. Therefore, we have:

Proposition 4.3**.**

Fix $\displaystyle z\in\mathbb{C}$ and $\displaystyle b\in(0,\infty)$. Then $\displaystyle\nu\in\operatorname{Sp}{L_{\text{PC}}\left(z,b\right)}$ if and only if

[TABLE]

Remark 4.4*.*

This result is a slight generalization of [Dem05, Eq. 8, p. 1083]. To see the compatibility, take (4.15), divide by $\displaystyle\frac{\sqrt{\pi}}{\sqrt{2}\Gamma(-\nu)}$, let $\displaystyle z=iV$, rename $\displaystyle b$ as $\displaystyle z_{1}$ and $\displaystyle\nu$ as $\displaystyle\xi$, and use Fact 2.2 to rewrite the product of the odd and even solutions as a power of $\displaystyle 2$, some Gamma functions, and $\displaystyle D_{\xi}^{2}(-z_{1})-D_{\xi}^{2}(z_{1})$.

4.3. Separation of Variables

We separate the variables, at the cost of making some functions in the equation meromorphic in $\displaystyle\nu$.

Corollary 4.5**.**

Fix $\displaystyle z\in\mathbb{C}$ and $\displaystyle b\in(0,\infty)$. For $\displaystyle\nu\in\mathbb{C}\setminus\mathbb{N}_{0}$, $\displaystyle\nu\in\operatorname{Sp}{L_{\text{PC}}\left(z,b\right)}$ if and only if

[TABLE]

where

[TABLE]

The pole of $\displaystyle M(\nu;b)$ at $\displaystyle\nu=0$ is not removable, so the function $\displaystyle M(\cdot;b)$ is not constant in $\displaystyle\nu$. Also, $\displaystyle M(\nu;b)$ is real if $\displaystyle\nu$ is real.

Proof.

For $\displaystyle\nu\not\in\mathbb{N}_{0}$, transforming (4.15) to (4.16) is elementary algebra. To demonstrate that the pole of $\displaystyle M(\nu,b)$ at $\displaystyle\nu=0$ is not removable, we need to prove that $\displaystyle D_{0}(b)$, $\displaystyle y_{\textnormal{even}}(0;b)$, and $\displaystyle y_{\textnormal{odd}}(0;b)$ are not $\displaystyle 0$.

$\displaystyle D_{0}(b)\neq 0$**: **

It is known (e.g, [MOS66, Section 8.1.2, p. 326]) that $\displaystyle D_{0}(b)=\exp\left(-\frac{b^{2}}{4}\right)$, and this is nonzero.

$\displaystyle y_{\textnormal{even}}(0;b)\neq 0$**: **

By (\theparentequation.i),

[TABLE]

and this is nonzero.

$\displaystyle y_{\textnormal{odd}}(0;b)\neq 0$**: **

By $\displaystyle\eqref{eq:yOddDirect}$ and $\displaystyle b>0$,

[TABLE]

Since the pole at $\displaystyle\nu=0$ is not removable, we must have that $\displaystyle\lim_{\nu\to 0}\left|M(\nu;b)\right|=\infty$, and hence $\displaystyle M(\nu;b)$ is non-constant in $\displaystyle\nu$.

The reality of $\displaystyle y_{\textnormal{even}}(\nu;b)$ and $\displaystyle y_{\textnormal{odd}}(\nu;b)$ if $\displaystyle\nu$ is real comes from the power-series expansions (3.7), as all summands are real. Then the decomposition of $\displaystyle D_{\nu}(x)$ in terms of the even and odd solutions, i.e. (3.9), and the reality of the gamma function for real inputs in its domain, ensures that $\displaystyle D_{\nu}(b)$ is real for real inputs. ∎

4.4. Notation

So far, $\displaystyle x$ has been the key variable, as our functions were functions of $\displaystyle x$, and all derivatives in (\theparentequation.i) are in $\displaystyle x$. Now that the eigenvalue equation has been formed, the emphasis shifts, and $\displaystyle\nu$ becomes the primary variable in the sequel. We choose a fixed $\displaystyle b>0$ and suppress explicit references to $\displaystyle b$:

[TABLE]

Thus, we rewrite Proposition 4.3 and Corollary 4.5 as follows.

Corollary 4.6**.**

Fix $\displaystyle z\in\mathbb{C}$. $\displaystyle\nu\in\mathbb{C}$ is in $\displaystyle\operatorname{Sp}{L_{\text{PC}}\left(z,b\right)}$ if and only if

[TABLE]

If, in addition, $\displaystyle\nu\in\mathbb{C}\setminus\mathbb{N}_{0}$, then $\displaystyle\nu\in\operatorname{Sp}{L_{\text{PC}}\left(z,b\right)}$ if and only if

[TABLE]

where

[TABLE]

The pole of $\displaystyle\mathsf{M}$ at $\displaystyle 0$ is never removable, so the function $\displaystyle\mathsf{M}(\nu)$ is not constant. Also, $\displaystyle\mathsf{M}$ is real-valued for $\displaystyle\nu$ real.

5. Zeros of Parabolic Cylinder Functions in the Parameter

An observation from (4.20) is that as $\displaystyle\left|z\right|\to\infty$, $\displaystyle\left|\frac{1}{z^{2}}\right|\to 0$, so the zeroes of $\displaystyle\mathsf{M}(\nu)$ outside $\displaystyle\mathbb{N}_{0}$, in particular of its factor $\displaystyle\mathsf{D}(\nu)$ (see (4.21)), are helpful in discerning the asymptotic behavior of the eigenvalues as $\displaystyle\left|z\right|$ grows large. For numerical confirmation of this idea, see Figure 1.

To ensure that we have zeroes of $\displaystyle\mathsf{D}(\nu)$ to work with, we prove:

Proposition 5.1**.**

$\displaystyle\mathsf{D}(\nu)$* has infinitely many distinct zeroes.*

The statement is implied by Figure 2 (p. 280) of Dean’s paper [Dea66], but we prefer to give a proof. Our approach uses the theory of entire functions, in particular the concept of (exponential) order of an entire function.

Definition 5.2**.**

Let $\displaystyle f(z)$ be an entire function. Then $\displaystyle f$ is of finite (exponential) order if there exists $\displaystyle 0<\rho<\infty$ such that for all $\displaystyle r\geq R=R(\rho)$,

[TABLE]

If such a $\displaystyle\rho$ exists, we call the infimum of such $\displaystyle\rho$ the exponential order (or simply order) of $\displaystyle f$.

For an entire function $\displaystyle f(\nu)$, define the maximum function $\displaystyle M_{f}(r)$, $\displaystyle r>0$, by

[TABLE]

by the Maximum Modulus Principle, $\displaystyle M_{f}(r)=\sup_{\left|\nu\right|\leq r}\left|f(\nu)\right|$.

Remark 5.3*.*

It makes little difference to our entire-functions arguments whether the suppressed argument $\displaystyle b$ is real or complex; therefore, for this section only, we will let $\displaystyle\beta$ be an arbitrary complex number and redefine $\displaystyle\mathsf{D}(\nu)=D_{\nu}(\beta)$; in this more general setting, the function is still at most order-$\displaystyle 1$, maximal-type in $\displaystyle\nu$.

For the moment, we assuming the following fact about the growth rate $\displaystyle\mathsf{D}(\nu)$, to be proven later.

Lemma 5.4**.**

$\displaystyle\mathsf{D}(\nu)$* is of exponential order at most $\displaystyle 1$ in $\displaystyle\nu$, though possibly of maximal type; more specifically, we have the estimate,*

[TABLE]

We also require the fact that $\displaystyle\mathsf{D}(\nu)$ decays as $\displaystyle\nu\to-\infty$.

Fact 5.5** (e.g., [MOS66, Chapter 8, Section 8.1.6, p. 332]).**

If $\displaystyle\displaystyle\left|\arg(-\nu)\right|\leq\frac{\pi}{2}$,

[TABLE]

In particular, for $\displaystyle\nu=-\xi$, $\displaystyle\xi>0$, we have as $\displaystyle\xi\to+\infty$ that

[TABLE]

Proof of Proposition 5.1, given Lemma 5.4.

Suppose, by way of contradiction, that $\displaystyle\mathsf{D}(\nu)$ has only finitely many zeros. Since $\displaystyle\mathsf{D}(\nu)$ is an entire function by the Weierstrass Factorization Theorem (e.g., [Lev64, Chapter 1, Section 3, Theorem 3, p. 8]), we must have

[TABLE]

Yet $\displaystyle\mathsf{D}(\nu)$ is a function of order at most $\displaystyle 1$ by Lemma 5.4, so $\displaystyle g(\nu)$ must be a degree-$\displaystyle 1$ polynomial; else, it would not be order $\displaystyle 1$. Therefore,

[TABLE]

Let $\displaystyle M$ be the maximum modulus of the zeroes of $\displaystyle P(\nu)$, i.e., of $\displaystyle\mathsf{D}(\nu)$. Then for $\displaystyle\nu=-\xi$, $\displaystyle\xi>M$, we must have $\displaystyle\mathsf{D}(-\xi)\neq 0$ and hence

[TABLE]

Yet by Fact 5.5, (5.4), $\displaystyle\ln\left|\mathsf{D}(-\xi)\right|=-\frac{\xi}{2}\log(\xi)+\frac{\xi}{2}-O_{\beta}(\sqrt{\xi})$ as $\displaystyle\xi\to+\infty$. Contradiction.

Therefore, $\displaystyle\mathsf{D}(\nu)$ has infinitely many zeros. ∎

Proof of Lemma 5.4.

We have (see [Olv61, Section 3, Theorem, (3.1), p. 813–4] ) that if $\displaystyle\nu=-a-\frac{1}{2}=\mu^{2}-\frac{1}{2}$, $\displaystyle\beta=\mu t\sqrt{2}$, and with $\displaystyle\theta=\arg\mu$, $\displaystyle\left|\theta\right|\leq\frac{\pi}{2}$,111This range of $\displaystyle\theta$ suffices, as $\displaystyle\left|\arg\mu\right|\leq\frac{\pi}{2}$ implies $\displaystyle\left|\arg(\mu^{2})\right|\leq\pi$, and $\displaystyle\nu=\mu^{2}+\frac{1}{2}$; hence, all (sufficiently large) $\displaystyle\nu$ may be achieved by rewriting in this way.

[TABLE]

where $\displaystyle k$ is an unspecified constant, the powers $\displaystyle(2e)^{-\,\frac{1}{4}\mu^{2}}$ and $\displaystyle\mu^{\frac{1}{2}\mu^{2}}$ have their principal values, and $\displaystyle\xi_{\theta}(t)$ is a particular branch of the multivalued function

[TABLE]

which is defined in the following way. As we are interested in the growth rate as $\displaystyle\left|\nu\right|\to\infty$, hence $\displaystyle\left|\mu\right|=\sqrt{\left|\nu+\frac{1}{2}\right|}\to\infty$, and $\displaystyle\beta$ is fixed, we will have $\displaystyle t=\frac{x}{\mu\sqrt{2}}\to 0$. For $\displaystyle\left|t\right|\leq\frac{1}{4}$, the correct branches for our purposes are ([Olv59, Section 5, (5.8 – 5.9), p. 143], [Olv61, Section 2, (2.8 – 2.10), p. 813]),

[TABLE]

More specifically, the requirements are that ([Olv61, Section 2, (2.9 – 2.10), p. 813])

[TABLE]

Since $\displaystyle\eta(0)=\frac{\pi}{4}$, it follows that for $\displaystyle\theta=0$, $\displaystyle\xi_{0}(t)=i\eta(t)$ for $\displaystyle\operatorname{Im}t<0$ and $\displaystyle t$ small, and $\displaystyle\xi_{0}(t)=-i\eta(t)$ for $\displaystyle\operatorname{Im}t>0$ and $\displaystyle t$ small. As $\displaystyle\theta$ changes, the branch cut moves, but wherever it is, below the branch cut, we use $\displaystyle\xi_{\theta}(t)=i\eta(t)$, and above it we use $\displaystyle\xi_{\theta}(t)=-i\eta(t)$, and on it, we will use whichever branch gives the larger upper bound for $\displaystyle\left|\exp(-\mu^{2}\xi_{\theta}(t))\right|$.

To ensure that $\displaystyle\left|t\right|\leq\frac{1}{4}$ above, it suffices to have $\displaystyle\left|\nu\right|\geq 8\left|\beta\right|^{2}+\frac{1}{2}$, since $\displaystyle\mu^{2}=\nu+\frac{1}{2}$, or $\displaystyle\left|\mu^{2}\right|\geq\left|\nu\right|-\frac{1}{2}\geq 8\left|\beta\right|^{2}$, or $\displaystyle\left|\mu\right|\geq 2\sqrt{2}\left|\beta\right|$, and then

[TABLE]

With $\displaystyle\left|t\right|\leq\frac{1}{4}$, we have that $\displaystyle\frac{\sqrt{15}}{4}\leq\left|\sqrt{1-t^{2}}\right|\leq 1$ and

[TABLE]

so that (recalling that $\displaystyle\mu^{2}=\nu+\frac{1}{2}$)

[TABLE]

Thus,

[TABLE]

To bound the other terms in the numerator of (5.7), we have

[TABLE]

and

[TABLE]

Therefore, combining (5.7), (5.8), (5.9), and (5.10),

[TABLE]

∎

6. Eigenvalue Localization Around Non-integer Roots of $\displaystyle\mathsf{D}$

6.1. Preliminaries

We now reformulate the the first part of Lemma  3.3 as

Lemma 6.1**.**

If $\displaystyle\nu\not\in\mathbb{N}_{0}$ and $\displaystyle\mathsf{D}(\nu)=0$ , then $\displaystyle\Phi(\nu)\neq 0$.

As $\displaystyle\mathsf{M}(\nu)=\Gamma(-\nu)[\mathsf{D}(\nu)]^{2}\Phi(\nu)$ (see (4.21)), zeroes of $\displaystyle\mathsf{D}(\nu)$ outside $\displaystyle\mathbb{N}$ result in zeroes of exactly double the order for $\displaystyle\mathsf{M}(\nu)$, and hence should be close to solutions of $\displaystyle\mathsf{M}(\nu)=\frac{1}{z^{2}}$ for $\displaystyle\left|z\right|$ large. We compress the complex analysis into the statements below. Let $\displaystyle\lambda\in\mathbb{C}\setminus\mathbb{N}$ be a zero of order exactly $\displaystyle m$ of $\displaystyle\mathsf{D}$. Then $\displaystyle\lambda$ is a zero of order exactly $\displaystyle 2m$ of $\displaystyle\mathsf{M}$; write this function locally as

[TABLE]

where $\displaystyle g(0)=0$ and $\displaystyle g$ is analytic in some disk; fix $\displaystyle R=R(\lambda)>0$ such that $\displaystyle\left|g(\zeta)\right|\leq\frac{1}{4}$ whenever $\displaystyle\left|\zeta\right|\leq R$. Let $\displaystyle B$ be the maximum of $\displaystyle\left|g^{\prime}(\zeta)\right|$ on this neighborhood.

Lemma 6.2**.**

Fix $\displaystyle\rho>0$. There exists a constant $\displaystyle\delta=\delta(\lambda,\rho)$ such that if $\displaystyle\left|z\right|>\frac{1}{\delta}$, there exist $\displaystyle 2m$ solutions of $\displaystyle\mathsf{M}(\nu)=\frac{1}{z^{2}}$ (i.e., (4.20)), within the radius-$\displaystyle\rho$-neighborhood of $\displaystyle\lambda$; more precisely, if $\displaystyle\epsilon\in\mathbb{C}$ is defined such that $\displaystyle\epsilon^{2m}=\frac{1}{z^{2}c_{2m}}$, then the $\displaystyle 2m$ solutions are

[TABLE]

In particular,

[TABLE]

Proof.

Keeping in mind (6.1), and letting $\displaystyle z=\nu-\lambda$, let us analyze an equation

[TABLE]

Let

[TABLE]

Put

[TABLE]

Then

[TABLE]

If $\displaystyle\omega^{p}=1$, $\displaystyle\omega=\exp\left(\frac{2\pi i}{p}\right)$, the equation (\theparentequation.i) splits into a series of $\displaystyle p$ equations

[TABLE]

where $\displaystyle c^{1/p}$ is any $\displaystyle p$th root of $\displaystyle c$. Each of them, for small enough $\displaystyle w$, has one and only one solution $\displaystyle z_{j}$, as it follows from analysis of the inverse function $\displaystyle z(\zeta)$ for

[TABLE]

it is well-defined for small $\displaystyle z$ because

[TABLE]

But we need good inequalities. By (6.9)

[TABLE]

i.e.,

[TABLE]

Claim 6.3**.**

If

[TABLE]

then

[TABLE]

Proof.

Indeed, $\displaystyle e^{1/3}<\frac{3}{2}$, so

[TABLE]

and for $\displaystyle z,t\in\mathbb{D}_{\kappa}$, by (\theparentequation.iii), (6.12)

[TABLE]

∎

The Claim is proven. It implies the existence of $\displaystyle z(\zeta)\in\eqref{eq:fixpoint}$ which gives solutions of (6.8)

[TABLE]

Additional properties of $\displaystyle z(\zeta)$ come from (6.10) — and (6.14), (\theparentequation.ii)

[TABLE]

For solutions (6.16), we have if

[TABLE]

[TABLE]

So far, (6.3) has not been explained, but if $\displaystyle\left|\epsilon\right|\leq\min\{\frac{1}{6B},\frac{\rho}{3}\}$, then $\displaystyle\left|E_{j}\right|\leq 3B\left(\frac{1}{6B}\right)\left|\epsilon\right|=\frac{\left|\epsilon\right|}{2}$, and hence

[TABLE]

and

[TABLE]

This explains that we may take $\displaystyle\delta=\sqrt{\left|c\right|\cdot\min\{\frac{1}{(6B)^{p}},\frac{1}{\left|c\right|[6(1+B)]^{p}},\left(\frac{\rho}{2}\right)^{p}\}}$. ∎

6.2. Case: real perturbation

Suppose $\displaystyle z=r$ is real in (1.8), i.e.,

[TABLE]

Lemma 6.4**.**

For $\displaystyle r\in\mathbb{R}$, $\displaystyle L_{\text{PC}}\left(r,b\right)$ is self-adjoint; consequently, $\displaystyle\operatorname{Sp}{L_{\text{PC}}\left(r,b\right)}\subseteq\mathbb{R}$.

Proof.

For real $\displaystyle r$, we now show that the quadratic form from which $\displaystyle L_{\text{PC}}\left(r,b\right)$ is formed, i.e.,

[TABLE]

is semi-bounded below, i.e.,

[TABLE]

Indeed, for all $\displaystyle f\in\mathcal{H}^{1}(\mathbb{R})$, for all $\displaystyle\epsilon>0$, there exists $\displaystyle T=T(\epsilon)$ such that

[TABLE]

For $\displaystyle r=0$,

[TABLE]

For $\displaystyle r\neq 0$, we let $\displaystyle\epsilon=\frac{1}{2\sqrt{\left|r\right|}}$ and apply the inequalities (6.21) and $\displaystyle(\alpha+\beta)^{2}\leq 2(\alpha^{2}+\beta^{2})$. Then

[TABLE]

and similarly for $\displaystyle-ru(-b)\overline{v(-b)}$. Therefore,

[TABLE]

Hence, in all cases, $\displaystyle\mathfrak{t}_{r,b}$ is semibounded below; as in [MS16], one checks that it is closed. Hence, the operator coming from the quadratic form is self-adjoint (e.g., [RS72, Chapter VIII, Section 6, Theorem VIII.15, p. 279]). ∎

Proposition 6.5**.**

1. (i)

If $\displaystyle\lambda\in\mathbb{C}$ and $\displaystyle\mathsf{D}(\lambda)=0$, then $\displaystyle\lambda>0$. 2. (ii)

If $\displaystyle\lambda\in\mathbb{R}\setminus\mathbb{N}$ is a zero of $\displaystyle\mathsf{D}$, it is a simple zero. 3. (iii)

If $\displaystyle\lambda\in\mathbb{R}\setminus\mathbb{N}$ is a zero of $\displaystyle\mathsf{D}$, then in the local series expansion

[TABLE]

$\displaystyle c_{2}>0$.

Proof, Part (i).

Fix $\displaystyle\lambda$ such that $\displaystyle\mathsf{D}(\lambda)=0$. If $\displaystyle\lambda\in\mathbb{N}$, we know that $\displaystyle\lambda$ is real, and we know that $\displaystyle\mathsf{D}(0)=D_{0}(b)=e^{-b^{2}/4}$ is nonzero, so we reduce to the case where $\displaystyle\lambda\in\mathbb{C}\setminus\mathbb{N}_{0}$. Fix $\displaystyle\rho>0$. By Lemma 6.2, for $\displaystyle r>0$ large enough such that $\displaystyle\frac{1}{r}<\delta(\lambda,\rho)$, there exist several solutions of $\displaystyle\mathsf{M}(\nu)=\frac{1}{r^{2}}$ within $\displaystyle\rho$ of $\displaystyle\lambda$. Yet these solutions cannot be nonnegative integers, for $\displaystyle\mathsf{M}(\nu)$ is not defined on the nonnegative integers. Therefore, by Corollary 4.6, these solutions are eigenvalues of $\displaystyle L_{\text{PC}}\left(r,b\right)$, and by Lemma 6.4, such eigenvalues must be real numbers. Therefore, we have that the distance from $\displaystyle\lambda$ to the real line is bounded above by the distance from $\displaystyle\lambda$ to $\displaystyle\operatorname{Sp}{L_{\text{PC}}\left(r,b\right)}$, which is at most $\displaystyle\rho$ for sufficiently large $\displaystyle r$. As $\displaystyle\rho>0$ is arbitrary, we must have $\displaystyle\lambda\in\mathbb{R}$.

Moreover, by the integral representation from (3.4) (e.g., [MOS66, Section 8.1.4, p. 328]), if $\displaystyle\nu<0$, then

[TABLE]

which is a nonzero constant times the integral of a positive integrand, and hence is nonzero. Therefore, the zeroes of $\displaystyle\mathsf{D}(\nu)$ avoid the negative real axis, and as already noted, $\displaystyle 0$ is not a zero of $\displaystyle\mathsf{D}(\nu)$, so $\displaystyle\lambda>0$. ∎

Proof, Part (ii).

Assume $\displaystyle m>1$, $\displaystyle 2m>2$. By Lemma 6.2, see (4.20), even if $\displaystyle z$ is real, with

[TABLE]

[TABLE]

($\displaystyle c_{2m}^{-1/2m}$ is any $\displaystyle 2m$th root of $\displaystyle c_{2m}^{-1}$), we have eigenvalues

[TABLE]

Therefore,

[TABLE]

and for $\displaystyle\epsilon$, $\displaystyle\left|\epsilon\right|<\frac{1}{12B}\cdot\frac{\pi}{m}$, at most $\displaystyle 2$ numbers in $\displaystyle\{\nu_{j}\}_{j=0}^{2m-1}$, $\displaystyle\nu_{j}\in\eqref{eq:localrootexpansionredo}$, are real. With $\displaystyle 2m\geq 4>2$ we would get a non-real eigenvalue of the self-adjoint operator $\displaystyle L_{\text{PC}}\left(z,b\right)$. This contradiction implies that $\displaystyle m=1$. ∎

Proof, Part (iii).

Moreover, the same analysis shows that for $\displaystyle m=1$ the coefficient $\displaystyle c_{2}\neq 0$ should be positive: $\displaystyle c_{2}>0$. Otherwise (i.e., with $\displaystyle c_{2}<0$), by (6.23) with positive $\displaystyle z>0$, $\displaystyle z\gg 1$,

[TABLE]

By (6.24)

[TABLE]

where

[TABLE]

For $\displaystyle z>1$ large enough, both eigenvalues $\displaystyle\nu_{0}$, $\displaystyle\nu_{1}$ would not be real.

∎

6.3. Case: Pure Imaginary perturbation

We now discuss $\displaystyle L_{\text{PC}}\left(ir,b\right)$, $\displaystyle r\in\mathbb{R}$, $\displaystyle b>0$, as in (1.8).

Proposition 6.6**.**

Fix $\displaystyle\lambda\in\mathbb{R}_{>0}\setminus\mathbb{N}_{0}$ such that $\displaystyle\mathsf{D}(\lambda)=0$, and fix $\displaystyle\rho>0$. For $\displaystyle r>\frac{1}{\delta(\lambda,\rho)}$, there exists $\displaystyle 2$ nonreal eigenvalues of $\displaystyle L_{\text{PC}}\left(ir,b\right)$ in a radius-$\displaystyle\rho$ neighborhood of $\displaystyle\lambda$.

Proof.

We repeat the above analysis. Let $\displaystyle z=ir$; then (6.26) becomes

[TABLE]

Then (6.27) and (6.28) still hold; hence, for $\displaystyle r$ large enough, the solutions of $\displaystyle\mathsf{M}(\nu)=\frac{1}{(ir)^{2}}$ near $\displaystyle\lambda$ are non-real. Yet by Corollary 4.6, these solutions are eigenvalues of $\displaystyle L_{\text{PC}}\left(ir,b\right)$. By Lemma 6.2, for $\displaystyle r$ large enough, the non-real eigenvalues are within $\displaystyle\rho$ of $\displaystyle\lambda$. ∎

7. Eigenvalue Localization around Integer Zeros of $\displaystyle\mathsf{D}$

7.1. One Guaranteed Zero, and Several Nearby Zeros

Although $\displaystyle D_{0}(x)=e^{-x^{2}/4}$ is never zero, it is certainly possible that $\displaystyle D_{n}(b)=0$ for $\displaystyle n\in\mathbb{N}$; indeed, $\displaystyle D_{2}(1)=0$.

By Corollary 4.6, (4.19), $\displaystyle\nu\in\operatorname{Sp}{L_{\text{PC}}\left(z,b\right)}$ if and only if

[TABLE]

If $\displaystyle\mathsf{D}(n)=0$, then $\displaystyle n\in\operatorname{Sp}{L_{\text{PC}}\left(z,b\right)}=0$ for all $\displaystyle z$, for the left-hand-side of the above equation becomes

[TABLE]

Lemma 7.1**.**

If $\displaystyle n\in\mathbb{N}$ is a zero of $\displaystyle\mathsf{D}$, then $\displaystyle n\in\operatorname{Sp}{L_{\text{PC}}\left(z,b\right)}$ for all $\displaystyle z\in\mathbb{C}$.

We claim, however, that neighborhoods of such integer zeros of $\displaystyle\mathsf{D}$ work in much the same way as non-integer zeros of $\displaystyle\mathsf{D}$ as a starting point for finding non-real eigenvalues of $\displaystyle\operatorname{Sp}{L_{\text{PC}}\left(ir,b\right)}$.

Numerical evidence for this in the case $\displaystyle b=1$ is given by Figure 2.

The key observation is the second part of Lemma 3.3, revised here as

Lemma 7.2**.**

If $\displaystyle n\in\mathbb{N}$, and $\displaystyle\mathsf{D}(n)=0$, then $\displaystyle\Phi(n)=0$.

Thus, if $\displaystyle n\in\mathbb{N}$ is a zero of $\displaystyle\mathsf{D}$ of order $\displaystyle m\geq 1$, then $\displaystyle n$ is a zero of $\displaystyle\Phi$ of order $\displaystyle q\geq 1$. Then $\displaystyle[\mathsf{D}(\nu)]^{2}\Phi(\nu)$ has a zero of order at least $\displaystyle 2m+q\geq 3$ at $\displaystyle\nu=n$, and thus $\displaystyle\mathsf{M}(\nu)=\Gamma(-\nu)[\mathsf{D}(\nu)]^{2}\Phi(\nu)$ (see (4.21)) has a removable singularity at $\displaystyle n$ and can be analytically continued there, with a zero of order $\displaystyle 2m+q-1\geq 2$ at $\displaystyle n$. We call this extended function $\displaystyle\mathbf{M}(\nu)$ for clarity; write this function locally as

[TABLE]

where $\displaystyle g(0)=0$ and $\displaystyle g$ is analytic in some disk; fix $\displaystyle R>0$ such that $\displaystyle\left|g(\xi)\right|\leq\frac{1}{4}$ whenever $\displaystyle\left|\xi\right|\leq R$, and let $\displaystyle B$ the maximum of $\displaystyle g^{\prime}(\xi)$ on this interval. We now establish the analogue of Lemma 6.2 for $\displaystyle\mathbf{M}(\nu)$.

Lemma 7.3**.**

Fix $\displaystyle\rho$, $\displaystyle 0<\rho<\frac{1}{2}$, and suppose that $\displaystyle\mathsf{D}(n)=0$ for $\displaystyle n\in\mathbb{N}$. Then there exists a constant $\displaystyle\delta(n,\rho)$ such that if $\displaystyle\left|z\right|>\frac{1}{\delta(n)}$, there exist $\displaystyle 2m+q-1$ solutions of $\displaystyle\mathsf{M}(\nu)=\frac{1}{z^{2}}$ (i.e., (4.20)), within the radius - $\displaystyle\rho$ - neighborhood of $\displaystyle\lambda$; more precisely, if $\displaystyle\epsilon\in\mathbb{C}$ is defined such that $\displaystyle\epsilon^{2m+q-1}=\frac{1}{z^{2}c_{2m+q-1}}$, then the $\displaystyle 2m+q-1$ solutions are

[TABLE]

In particular,

[TABLE]

Proof.

By repeating the proof of Lemma 6.2, for sufficiently large $\displaystyle z$, there exist $\displaystyle 2m+q-1$ solutions of $\displaystyle\mathbf{M}(\nu)=\frac{1}{z^{2}}$ with the above properties. However, by the first inequality of (7.3), none of the solutions $\displaystyle\nu_{j}$ are equal to $\displaystyle n$ and by the latter two inequalities, $\displaystyle\left|\nu_{j}-n\right|<\rho<\frac{1}{2}<1$, no solution $\displaystyle\nu_{j}$ can equal any integer other than $\displaystyle n$. Therefore, no $\displaystyle\nu_{j}$ is integer, which means that the solutions are solutions of the unextended equation $\displaystyle\mathsf{M}(\nu)=\frac{1}{z^{2}}$ as well. ∎

7.2. Case: Real Perturbation

We now prove the analogue of Proposition 6.5 for integer zeros of $\displaystyle\mathsf{D}$.

Proposition 7.4**.**

1. (i)

If $\displaystyle n\in\mathbb{N}$ is a zero of $\displaystyle\mathsf{D}$ of order $\displaystyle m$, and it is a zero of $\displaystyle\Phi$ of order $\displaystyle q$, then both zeros are simple; i.e., $\displaystyle\mathbf{M}$ has a zero of order exactly $\displaystyle 2$ there. 2. (ii)

If $\displaystyle n\in\mathbb{N}$ is a zero of $\displaystyle\mathsf{D}$, then in the series expansion

[TABLE]

$\displaystyle c_{2}>0$.

Proof, part (i).

If $\displaystyle n\in\mathbb{N}$ is a zero of $\displaystyle\mathsf{D}$ of order $\displaystyle m$, and $\displaystyle n$ is a zero of $\displaystyle\Phi$ of order $\displaystyle q$, then the order of the zero of $\displaystyle\mathbf{M}$ at $\displaystyle n$ is $\displaystyle 2m+q-1$, since $\displaystyle[\mathsf{D}(\nu)]^{2}$ gives a zero of order $\displaystyle 2m$ at $\displaystyle n$, and $\displaystyle\Gamma(-\nu)$ takes away an order of the zero. Thus, let us write $\displaystyle\mathbf{M}$ locally as

[TABLE]

Let

[TABLE]

where $\displaystyle c_{2m+q-1}^{-1/(2m+q-1)}$ is any $\displaystyle 2m+q-1$th root of $\displaystyle c_{2m+q-1}^{-1}$. By Lemma 7.3, letting $\displaystyle z$ be real, there exist $\displaystyle 2m+q-1$ solutions of $\displaystyle\mathbf{M}(\nu)=\frac{1}{z^{2}}$ in a radius-$\displaystyle\frac{1}{2}$-neighborhood of $\displaystyle n$, namely

[TABLE]

where $\displaystyle\omega=\exp\left(\frac{2\pi}{2m+q-1}\right)$. By Corollary 4.6, these solutions are eigenvalues of $\displaystyle L_{\text{PC}}\left(z,b\right)$, and by Lemma 6.4, they are real.

Therefore,

[TABLE]

and for $\displaystyle\epsilon$ such that $\displaystyle\left|\epsilon\right|<\frac{1}{12B}\cdot\frac{2\pi}{2m+q-1}$, at most $\displaystyle 2$ numbers in $\displaystyle\{\nu_{j}\}_{j=0}^{2m+q-2}$, $\displaystyle\nu_{j}\in\eqref{eq:localsolInt}$, are real. With $\displaystyle 2m+q-1\geq 3>2$, we would get a non-real eigenvalue of the self-adjoint operator $\displaystyle L_{\text{PC}}\left(z,b\right)$. This contradiction implies that $\displaystyle 2m+q-1\leq 2$, and with $\displaystyle m\geq 1$, $\displaystyle q\geq 1$, the only resolution is $\displaystyle m=q=1$; i.e., both roots are simple. ∎

Proof, part (ii).

For $\displaystyle m=q=1$, the coefficient $\displaystyle c_{2}\neq 0$ should be positive: i.e., $\displaystyle c_{2}>0$. Otherwise, by (7.5), with positive $\displaystyle z$, $\displaystyle z\gg 1$,

[TABLE]

By (7.6),

[TABLE]

where

[TABLE]

For $\displaystyle z>1$ large enough, both solutions $\displaystyle\nu_{0}$, $\displaystyle\nu_{1}$ would not be real, but they are real, again by Corollary 4.6 and Lemma 6.4. ∎

7.3. Case: Imaginary Perturbation

Proposition 7.5**.**

If $\displaystyle n\in\mathbb{N}$ is such that $\displaystyle\mathsf{D}(n)=0$, and $\displaystyle\rho>0$, then for $\displaystyle r>\frac{1}{\delta(n,\rho)}$, there exists $\displaystyle 2$ nonreal eigenvalues of $\displaystyle L_{\text{PC}}\left(ir,b\right)$ (see (1.8)) in the radius-$\displaystyle\rho$-neighborhood of $\displaystyle n$.

Proof.

We repeat the above analysis. If $\displaystyle z=ir$, then (7.8) becomes

[TABLE]

Then (7.9) and (7.10) still hold; hence, for $\displaystyle r$ large enough, the solutions of $\displaystyle\mathsf{M}(\nu)=\frac{1}{(ir)^{2}}$ near $\displaystyle n$ are non-real. yet by Corollary 4.6, these solutions are eigenvalues of $\displaystyle L_{\text{PC}}\left(ir,b\right)$. ∎

8. Proof of Theorem 1

Consider the family of operators $\displaystyle L_{\text{PC}}\left(re^{i\pi/2},b\right)=L_{\text{PC}}\left(ir,b\right)$. Consider the first $\displaystyle p$ roots $\displaystyle 0<\lambda_{1}<\dotsc<\lambda_{p}$ of $\displaystyle\mathsf{D}(\nu)$. By Lemma 6.2 and Proposition 6.5 (or Lemma 7.3 and Proposition 7.4), for each $\displaystyle q$, $\displaystyle 1\leq q\leq p$, the equation $\displaystyle\mathsf{M}(\nu)=\frac{1}{z^{2}}$ (or $\displaystyle\mathbf{M}(\nu)=\frac{1}{z^{2}}$) is of the form

[TABLE]

where $\displaystyle g(z)\leq C_{3}$ if $\displaystyle\left|z\right|\leq\rho_{q}$, and we may enforce $\displaystyle\rho_{q}<\frac{1}{2}\min_{2\leq q\leq p}\lambda_{q}-\lambda_{q-1}$. By Propositions 6.6 and 7.5, for each $\displaystyle q$, $\displaystyle 1\leq q\leq p$, there exists a constant $\displaystyle\delta(\lambda_{q},\rho_{q})$ such that for $\displaystyle r>\frac{1}{\delta(\lambda_{q},\rho_{q})}$, there exist $\displaystyle 2$ non-real solutions of (8.1) in the radius-$\displaystyle\rho_{q}$-neighborhood of $\displaystyle\lambda_{q}$ (call them $\displaystyle\kappa_{j}^{(q)}(r)$), $\displaystyle j=0,1$), and by Corollary 4.6, these solutions are eigenvalues of $\displaystyle\operatorname{Sp}{L_{\text{PC}}\left(ir,b\right)}$. Moreover, by $\displaystyle\rho_{q}<\frac{1}{2}\min_{2\leq q\leq p}\lambda_{q}-\lambda_{q-1}$, these neighborhoods are disjoint, so if $\displaystyle q\neq t$, $\displaystyle\kappa_{j}^{(q)}(r)\neq\kappa_{k}^{(t)}(r)$, $\displaystyle j,k$ in $\displaystyle\{0,1\}$. Therefore, we see that for some $\displaystyle r^{*}$,

[TABLE]

This holds for all $\displaystyle p\in\mathbb{N}$, so letting

[TABLE]

we have the following.

Theorem 8.1**.**

$\displaystyle\lim_{r\to\infty}\mathcal{N}_{\textnormal{PC}}(r;b)=\infty$.

By the change of variables in Section 1.2, we establish Theorem 1, as well; i.e., $\displaystyle\lim_{r\to\infty}\mathcal{N}_{\textnormal{HO}}(r)=\infty$.

9. Conclusion

Two general questions should be clarified.

First, we know that

[TABLE]

But the gap between the estimates for $\displaystyle\mathcal{N}$ from above and below is too wide.

Second, how to the eigenvalues $\displaystyle\lambda(ir)$ move, $\displaystyle 0\leq r<\infty$? Sections 6, 7 tell a lot about $\displaystyle\lambda(ir)$, $\displaystyle r\gg 1$, close to zeros of $\displaystyle\mathsf{D}$. But, for example, could some $\displaystyle\lambda_{j}(ir)$, $\displaystyle 0\leq r\ll 1$, go to $\displaystyle\infty$ when $\displaystyle r\to\infty$? Numerics hint that it could not happen, but there is no formal (rigorous) argument to explain this phenomenon. Ran

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