Sequences of closely spaced resonances and eigenvalues for bipartite complex potentials

TL;DR

This paper analyzes the spectral properties of a Schroedinger operator with bipartite complex potentials separated by a large distance, revealing a sequence of resonances and eigenvalues near the real axis with potential applications in physics.

Contribution

It introduces a novel analysis of the eigenvalues and resonances for bipartite complex potentials with large separation, highlighting their approximate equidistance and physical relevance.

Findings

Existence of a sequence of approximately equidistant complex wavenumbers near the real axis.

Wavenumbers correspond to either resonances or eigenvalues depending on their imaginary parts.

Resonance sequence resembles transmission resonances in Fabry-Pérot interferometers.

Abstract

We consider a Schroedinger operator on the axis with a bipartite potential consisting of two compactly supported complex-valued functions, whose supports are separated by a large distance. We show that this operator possesses a sequence of approximately equidistant complex-valued wavenumbers situated near the real axis. Depending on its imaginary part, each wavenumber corresponds to either a resonance or an eigenvalue. The obtained sequence of wavenumbers resembles transmission resonances in electromagnetic Fabry-P\'erot interferometers formed by parallel mirrors. Our result has potential applications in standard and non-hermitian quantum mechanics, physics of waveguides, photonics, and in other areas where the Schroedinger operator emerges as an effective Hamiltonian.

Equations

$${\mathcal{H}}_{\ell }:=-\frac{d^2}{d{x}^2}+V_{\ell }(x),V_{\ell }(x):=V_{+}(x-\ell )+V_{-}(x+\ell ),$$

$$-{\psi }^{\prime \prime }+V_{\ell }(x)\psi =k^2\psi ,x\in \text{mathds}R,\psi (x)=C_{\pm }{e}^{\pm ikx},\pm x>x_0,C_{\pm }=\text{const},$$

$$-$$

$$Y_{\pm }(x,k)=e^{ikx},$$

$$F(k):=\frac{X_{-}^{\prime }(0,k)-ik{X}_{-}(0,k)}{X_{-}^{\prime }(0,k)+ik{X}_{-}(0,k)}\frac{X_{+}^{\prime }(d_{+},k)-ik{X}_{+}(d_{+},k)}{X_{+}^{\prime }(d_{+},-k)-ik{X}_{+}(d_{+},-k)}.$$

$$X_{-}^{\prime }(0,0)\ne 0,X_{+}^{\prime }(d_{+},0)\ne 0.$$

$$a_n:=\frac{\pi n}{2\ell },$$

$$\frac{e^{\frac{\pi }{2}}}{4\ell }\underset{\overline{B}}{\max }|F^{\prime }|<1,$$

$$k_{n}=a_{n}+\lim\limits_{m\to+\infty}h_{n}^{[m]}(0),\qquad h_{n}(k):=-\frac{i}{4\ell}\ln F(k+a_{n}),\qquad\big{|}k_{n}-a_{n}-h_{n}^{[m]}(0)\big{|}\leqslant\frac{\pi e^{\frac{m\pi}{2}}}{(4\ell)^{m+1}}{\max\limits_{\overline{B}}}^{m}|F^{\prime}|,$$

$$k_{n}=a_{n}+\sum\limits_{m=1}^{\infty}\frac{(-i)^{m}}{4^{m}m!\ell^{m}}\frac{d^{m-1}\ln^{m}F}{dk^{m-1}}\bigg{|}_{k=a_{n}}=\sum\limits_{m=1}^{\infty}\frac{1}{2^{m}m!\ell^{m}}\frac{d^{m-1}\ }{dk^{m-1}}\left(\pi n-\frac{i}{2}\ln F(k)\right)^{m}\bigg{|}_{k=0}.$$

$$\displaystyle\left|k_{n}-a_{n}-\sum\limits_{m=1}^{M}\frac{(-i)^{m}}{4^{m}m!\ell^{m}}\frac{d^{m-1}\ln^{m}F}{dk^{m-1}}\bigg{|}_{k=a_{n}}\right|\leqslant\frac{1}{4^{M+1}(M+1)!\ell^{M+1}}\max\limits_{\overline{B}}\left|\frac{d^{M}\ln^{M+1}F}{dk^{M}}\right|,$$

$$\displaystyle\left|k_{n}-\sum\limits_{m=1}^{M}\frac{1}{2^{m}m!\ell^{m}}\frac{d^{m-1}\ }{dk^{m-1}}\left(\pi n-\frac{i}{2}\ln F\right)^{m}\bigg{|}_{k=0}\right|\leqslant\frac{1}{2^{M+1}(M+1)!\ell^{M+1}}\max\limits_{\overline{B}}\left|\frac{d^{M}\ }{dk^{M}}\left(\pi n-\frac{i}{2}\ln F\right)^{M+1}\right|.$$

$$F(k)=F_{-}(k)F_{+}(k){\beta }_{-}^{-2}{\beta }_{+}^{-2},F_{\pm }(k)=2ik\sqrt{k^2+{\beta }_{\pm }^2}\cot \sqrt{k^2+{\beta }_{\pm }^2}+{\beta }_{\pm }^2+2k^2.$$

$$k_n\cong$$

$$F(k)=(2ik-{\beta }_{+})(2ik-{\beta }_{-}){\beta }_{-}^{-1}{\beta }_{+}^{-1}.$$

$$k_{n}\cong\frac{\pi n}{2\ell}-\frac{\pi n}{4\ell^{2}}\big{(}\beta_{+}^{-1}+\beta_{-}^{-1}\big{)}-\frac{1}{8\ell^{3}}\left(i\pi^{2}n^{2}\big{(}\beta_{+}^{-2}+\beta_{-}^{-2}\big{)}-\pi n\big{(}\beta_{+}^{-1}+\beta_{-}^{-1}\big{)}^{2}\right).$$

$$X_{\pm }(x,k)=e^{-ikx}+\underset{-\infty }{\overset{x}{\int }}\frac{\sin k(x-t)}{k}{V}_{\pm }(t)X_{\pm }(t,k)dt$$

$$|F(k)-1|<1-e^{-\frac{\pi }{2}}\text{as}k\in \overline{B}\Rightarrow e^{-\frac{\pi }{2}}<|F(k)|<2-e^{-\frac{\pi }{2}},k\in \overline{B}.$$

$$X_{\pm }(x,k)=a_{\pm }(k)Y_{\pm }(x,-k)+b_{\pm }(k)Y_{\pm }(x,k).$$

$$a_{-}(k)=\frac{ik{X}_{-}(0,k)-X_{-}^{\prime }(0,k)}{2ik},a_{+}(k)=\frac{ik{X}_{+}(d_{+},k)-X_{+}^{\prime }(d_{+},k)}{2ik{e}^{-ik{d}_{+}}},$$

$$b_{-}(k)=\frac{ik{X}_{-}(0,k)+X_{-}^{\prime }(0,k)}{2ik},b_{+}(k)=\frac{ik{X}_{+}(d_{+},k)+X_{+}^{\prime }(d_{+},k)}{2ik{e}^{ik{d}_{+}}}.$$

$$\psi (x,k)=a_{-}(k)e^{-ik(x+\ell )}+b_{-}(k)e^{ik(x+\ell )}=a_{-}(k)e^{-2ik\ell }{e}^{-ik(x-\ell )}+b_{-}(k)e^{2ik\ell }{e}^{ik(x-\ell )}.$$

$$\psi (x,k)=a_{-}(k)e^{-2ik\ell }{X}_{+}(x-\ell ,k)+b_{-}(k)e^{2ik\ell }{X}_{-}(x-\ell ,-k),x>\ell .$$

$$\psi (x,k)=$$

$$\displaystyle+{\Big{(}a_{-}(k)b_{+}(k)e^{-2ik\ell}+b_{-}(k)a_{-}(-k)e^{2ik\ell}\Big{)}e^{ik(x-\ell)}}.$$

$$e^{4ik\ell }=F(k).$$

$$z=h_n(z+a_n),k=z+a_n,h_n=-\frac{i}{4\ell }h.$$

$$|h(k)|=|\ln F(k)|⩽\sum _{m=1}^{\infty }\frac{|F(k)-1{|}^m}{m}⩽\sum _{m=1}^{\infty }\frac{(1-e^{-\frac{\pi }{2}}{)}^m}{m}=\frac{\pi }{2}$$

$$|h_{n}(z_{1})-h_{n}(z_{2})|=\left|\frac{i}{4\ell}\big{(}h(z_{1}+a_{n})-h(z_{2}+a_{n})\big{)}\right|=\frac{|F^{\prime}(z_{*}+a_{n})|}{4\ell|F(z_{*}+a_{n})|}|z_{1}-z_{2}|\leqslant\frac{e^{\frac{\pi}{2}}\max\limits_{\overline{B}}|F^{\prime}|}{4\ell}|z_{1}-z_{2}|,$$

$$\left|\frac{d^{m-1}{h}^m}{d{k}^{m-1}}(a_n)\right|⩽(m-1)!{\left(\frac{4\ell }{\pi }\right)}^{m-1}\underset{\overline{B}}{\max }|h^m|⩽\frac{\pi }{4\ell }(m-1)!(2\ell {)}^m,\left|\frac{(-i{)}^m}{4^{m}m!{\ell }^m}\frac{d^{m-1}{h}^m}{d{k}^{m-1}}(a_n)\right|⩽\frac{1}{2^{m}m}\frac{\pi }{4\ell },$$

$$h_n({\tilde{z}}_n+a_n)=$$

$$H_{m}^{(n)}=\frac{1}{m+1}\frac{d^{m}h^{m+1}}{dk^{m}}\quad\Leftrightarrow\quad\frac{1}{m+1}(\underbrace{h+\ldots+h}\limits_{m+1\ \text{times}})^{m}=\sum\limits_{\Theta\in\Pi_{m}}h^{|\Theta|}\prod\limits_{\theta\in\Theta}(|\theta||h|)^{|\theta|-1}=\sum\limits_{\Theta\in\Pi_{m}}h^{m}\prod\limits_{\theta\in\Theta}(|\theta|)^{|\theta|-1}$$

Full text

1 Introduction and motivation

Resonances play an important role in quantum mechanics [1] and in other branches of physics, where the interaction between a wave and a localized potential is described by an effective Schrödinger Hamiltonian. While the standard quantum mechanics deals with real-valued potentials, there is a steadily growing interest in non-hermitian quantum theories which involve complex potentials [2, 3, 4, 5]. Apart from this, non-self-adjoint Schrödinger Hamiltonians is a hot topic in many other fields, including theory of electromagnetic waveguides with gain and absorption [6], non-hermitian photonics [7], and parity-time-symmetric physics [8].

In this Letter, we consider the resonances and eigenvalues of a one-dimensional Schrödinger operator with a complex potential formed by two functions with compact supports separated by a sufficiently large distance. Let $V_{\pm}=V_{\pm}(x)$, $x\in\mathds{R}$, be two measurable bounded complex-valued functions and $\ell>0$ be a parameter. Suppose that the function $V_{+}$ is supported in $[0,d_{+}]$, while $V_{-}$ is supported in $[-d_{-},0]$, $d_{\pm}>0$, and introduce the Schrödinger operator

[TABLE]

on $\mathds{R}$. Rigorously we define $\mathcal{H}_{\ell}$ as an unbounded operator in $L_{2}(\mathds{R})$ on the domain $H^{2}(\mathds{R})$. This is an $m$-sectorial operator; if $V_{\pm}$ are real-valued functions, this operator is self-adjoint.

The main object of our study is the resonances and the eigenvalues of the operator $\mathcal{H}_{\ell}$. They correspond to $k\in\mathds{C}$, for which the problem

[TABLE]

possesses a non-trivial solution. Here $x_{0}$ is fixed number such that the function $V_{\ell}$ vanishes outside $[-x_{0},x_{0}]$, for instance, $x_{0}=\ell+d_{-}+d_{+}$, and the prime denotes the derivative with respect to $x$. Wavenumbers $k$ with $\operatorname{Im}k\leqslant 0$ describe the resonances $\lambda=k^{2}$ associated with non-$L_{2}$-integrable eigenfunctions $\psi$. Wavenumbers $k$ with $\operatorname{Im}k>0$ correspond to the eigenvalues $\lambda=k^{2}$ associated with integrable bound states $\psi$ having a finite $L_{2}$-norm.

2 Main result

Our main result states that as the distance $\ell$ is sufficiently large, problem (1) features a sequence of approximately equidistant values $k=k_{n}$ corresponding to resonances or eigenvalues. Proceeding to the rigorous formulation of the main result, we introduce some additional notations. By $X_{\pm}$, $Y_{\pm}$ we denote the solutions of the problems

[TABLE]

Next, we introduce a function

[TABLE]

By $\circ$ we denote the superposition of the functions, that is, $(f\circ g)(k)=f(g(k))$. An $m$-multiple superposition of a function $f$ is denoted by $f^{[m]}$, i.e., $f^{[m]}=\underbrace{f\circ f\circ\cdots\circ f}_{m\ \text{times}}$. The symbol $\lfloor\cdot\rfloor$ stands for the integer part of a number.

Our main result is the following theorem. Its proof is presented below in Section 5.

Theorem 1**.**

Assume that

[TABLE]

There exists $r>0$ depending on $V_{\pm}$ only, such that the function $F(k)$ is well-defined, holomorphic and non-zero on $B:=\{k\in\mathds{C}:\,|k|\leqslant r\}$. For all integer $n\in\mathds{Z}$ obeying $|n|\leqslant N_{\ell}$, $N_{\ell}:=\left\lfloor\frac{2\ell r}{\pi}-\frac{1}{2}\right\rfloor$, the circle B_{n}:=\big{\{}k\in\mathds{C}:\,|k-a_{n}|<\tfrac{\pi}{4\ell}\big{\}},

[TABLE]

possesses exactly one value $k_{n}(\ell)$, for which problem (1) has a non-trivial solution. As

[TABLE]

the value $k_{n}$ satisfies the representation

[TABLE]

where the branch of the logarithm is fixed by the condition $\arg\ln z\in(-\pi,\pi]$ and $m\in\mathds{N}$. The value $k_{n}$ can be also represented by absolutely uniformly in $\ell^{-1}$ convergent series

[TABLE]

For all $M\in\mathds{N}$ the inequalities hold:

[TABLE]

3 Discussion of the main result

Our main result states that as $\ell$ is not too small so that $N_{\ell}\geqslant 1$, there exist at least $2N_{\ell}+1$ complex-valued wavenumbers $k_{n}$, $|n|\leqslant N_{\ell}$, such that each $k_{n}$ corresponds to a nontrivial solution of problem (1) and therefore represents either a resonance or an eigenvalue of the operator $\mathcal{H}_{\ell}$. Values $k_{n}$ are located in small circles $B_{n}$ centered at the points $a_{n}$ in (4), one value in each circle, and form an approximately equidistant sequence situated close to the real axis. As $\ell$ increases, the number of the eigenvalues and resonances grows proportionally to $\ell$, while the distances between neighbouring wavenumbers $k_{n}$ tend to zero.

The obtained result is fairly general and holds independently of the specific shape of left ($V_{-}$) and right ($V_{+}$) components of the potential $V_{\ell}$, provided that $V_{\pm}$ have finite support and are separated well enough. Potential $V_{\ell}$ may be real or complex-valued, and therefore each $k_{n}$ represents either a resonance or an isolated (generically, complex) eigenvalue of non-self-adjoint operator $\mathcal{H}_{\ell}$. Nonzero values $k_{n}$ that eventually lie on the real axis correspond to spectral singularities (i.e., zero-width resonances). An important feature of our result is that it is valid not only for potentials $V_{\pm}$, but also for more general perturbations. Namely, we can replace the operators of multiplications by $V_{\pm}$ by more general operators but still acting on $[-d_{-},0]$ and $[0,d_{+}]$, and the statement of Theorem 1 remains the same. The main result is formulated only in terms of the function $F$ which is determined by the functions $X_{\pm}$ and their derivatives at certain points; no other information on $V_{\pm}$ is needed. This is why, for instance, we can assume that $V_{\pm}$ describe second-order differential operators, i.e. $V_{\pm}u=A_{\pm}^{(2)}u^{\prime\prime}+A_{\pm}^{(1)}u^{\prime}+A_{\pm}^{(0)}u$, where $A_{\pm}^{(j)}$ are smooth compactly supported on $[-d_{-},0]$ and $[0,d_{+}]$ complex functions. Another generalization for which Theorem 1 holds after some obvious modifications corresponds to a pair of delta-interactions $V_{\pm}=\beta_{\pm}\delta(x)$, where $\beta_{\pm}$ are complex constants (see Section 4).

A similar spectral picture was described in the recent study [9] devoted to the resonances of a one-dimensional discrete Schrödinger operator. The potential was of the form $V\chi_{L}$, where $V$ was either periodic or random, and $\chi_{L}$ was a characteristic function of the interval $[-L+1,L]$. It was shown, that as $L\to+\infty$, there was a sequence of closely spaced resonances accumulated along some analytic curve for the periodic potential or even a cloud of closely spaced resonances located in some domain for the random potential. Here the points $-L+1$ and $L$, at which the potential $V$ was replaced by zero, played the same role as our potentials $V_{\pm}$, and this explains the similarity of our spectral picture with that in [9]. The main result in [9] for periodic $V$ provided the existence of the resonances, the leading terms of their asymptotics and asymptotic description of the distribution density of the resonances. The same continuous model for the Schrödinger operator was treated in [10]. It was shown that there existed a sequence of closely spaced of resonances accumulated along a certain curve. We also note that in [18, Ch. I​I.1, Sect. I​I.1.4, Thm. 1.4.1], the operator $-\Delta+V_{1}+V_{2}(\cdot-\ell y)$ in $\mathds{R}^{3}$ was considered, where $V_{1,2}$ were real compactly supported functions and $y$ was a fixed point. It was proved that this operator possessed an infinite sequence of resonances converging to zero as $\ell\to\infty$, and their asymptotic behaviors were $k_{n}=\ell^{-1}k_{0,n}+\ell^{-2}k_{1,n}+o(\ell^{-2})$, where $ik_{0,n}|y|$ were the roots of the equation $z=\pm e^{z}$. Since the error term in this asymptotic law was non-uniform in $n$, this expansion was useful only for sufficiently large $\ell$ and some fixed $n$ and could not be effectively applied as $n\sim\ell$. No other information on the resonances was obtained in the cited book, so, this result just indicated the existence of some sequence of resonances without careful studying its properties. As we shall discuss below, our result provides much more information about the location of the resonances and, moreover, in our case there can be also ladders of eigenvalues or mixed ladders of coexisting eigenvalues and resonances since our potentials $V_{\pm}$ are not assume to be real-valued.

In our model, the sequence of eigenvalues and resonances emerges due to the large distance between the supports of $V_{\pm}$. The Schrödinger operators with several potentials separated by a large distance is a classical subject [13, 14, 15, 16]. The results of [16] applied to our operator $\mathcal{H}_{\ell}$ state that as $\ell\to+\infty$, the resolvent $(\mathcal{H}_{\ell}-\lambda)^{-1}$ splits into a direct sum of three resolvents $(\mathcal{H}_{\pm}-\lambda)^{-1}$ and $(\mathcal{H}_{0}-\lambda)^{-1}$, where $\mathcal{H}_{\pm}:=-\frac{d^{2}\ }{dx^{2}}+V_{\pm}$, $\mathcal{H}_{0}:=-\frac{d^{2}\ }{dx^{2}}$. The spectrum of $\mathcal{H}_{\ell}$ converges to the union of the spectra of $\mathcal{H}_{\pm}$ and $\mathcal{H}_{0}$. The essential spectrum of each of these operators is $[0,+\infty)$ and contains no embedded eigenvalues. In the vicinity of zero, the operators $\mathcal{H}_{\pm}$ and $\mathcal{H}_{0}$ can have only finitely many resonances. In other words, the operators $\mathcal{H}_{\pm}$ and $\mathcal{H}_{0}$ have only finitely many singularities of a (meromorphic continuation of) the resolvent. Despite this fact, as the distance between the supports of $V_{\pm}$ is large enough, the sequence of closely spaced eigenvalues and resonances of the operator $\mathcal{H}_{\ell}$ emerges.

Turning to the physical interpretation of our effect, it is relevant to point out that the constructed sequence of wave numbers resembles resonances emerging in optical Fabry-Pérot interferometers formed by two parallel plates [11]. If light is launched in such a resonator under normal incidence, then the constructive interference between internal reflections of light traveling between the plates results in a sequence of transmission resonances at wave vectors $k_{n}^{(FP)}=n\pi/L^{(FP)}$, where $L^{(FP)}$ is the geometrical length of the resonator. Considering the components of the bipartite potential as the “plates” of the resonator and noting that for large $\ell\gg 1$ the distance between the “plates” can be interpreted as the length of the resonator, i.e., $L^{(FP)}=2\ell$, we observe that the obtained expressions for the Fabry-Pérot resonances $k_{n}^{(FP)}=(n\pi)/(2\ell)$ coincide with the ball centers $a_{n}$ in (4) that give the location of our wavenumbers in the leading order. Therefore the obtained result can be interpreted as a Fabry-Pérot interferometer for waves governed by by effective Schrödinger-type Hamiltonians. This result extends the well-known analogy between the Fabry-Pérot interferometer and resonances of a quantum particle scattered by a single potential well or a barrier [12].

Our sequences of resonances and eigenvalues could also be a reflection of some kind of tunneling between the potentials $V_{\pm}$. At the same time, usually, the tunneling leads to an exponentially small asymptotic law for the eigenvalues, even in a very general case [17], while our resonances and eigenvalues exhibits power-law asymptotic behavior, see (7). The exponential asymptotic laws for usual tunneling are due to the exponential fall-off of the eigenfunctions for each single potential, while for the resonances the situation can be different and one can face an asymptotic power law. However, our sequence can also include eigenvalues.

Apart of the existence of the resonance and eigenvalues, Theorem 1 also provides two recipes for finding explicitly the corresponding wavenumbers $k_{n}(\ell)$. The first of them is the formula in (6) defining a convergent iterative process $k_{n}^{(m)}:=a_{n}+\eta^{(m)}_{n}$, where $m\in\mathds{N}$ counts the iterations, and $\eta^{(m)}_{n}$ is obtained recurrently as $\eta^{(m+1)}_{n}:=(4i\ell)^{-1}\ln F(\eta^{(m)}_{n}+a_{n})$, where $\eta^{(1)}_{n}:=(4i\ell)^{-1}\ln F(a_{n})$. This approach is perfectly suited for the numerical finding of $k_{n}$. The estimate in (6) controls the error between the exact value $k_{n}$ and the approximation $k_{n}^{(m)}$ after $m$ iterations; this estimate is independent of $n$. The radius $r$ of the circle $B$ is some implicit characteristics of the potentials $V_{\pm}$. As we shall show in the proof of Theorem 1, this radius must be chosen such that $|F(k)-1|<1-e^{-\frac{\pi}{2}}\approx 0.792$ as $|k|\leqslant r$.

Another approach to finding $k_{n}$ is provided by two absolutely uniformly convergent series (7) which should be regarded as Taylor series for $k_{n}$ in powers of $-\frac{i}{4\ell}$ either at zero or at the point $a_{n}$, no matter that $a_{n}$ depends on $\ell$ as well. The partial sums of these series approximate $k_{n}$ according to (8) and (9). We stress that the right-hand side of the inequality (8) is independent of $n$. The second series in equation (7) is well-adapted for providing asymptotic behavior for $k_{n}$ as $\ell\to\infty$. Despite now the coefficients are independent of $a_{n}$, the price we have to pay is a worse error term in (9): now it depends on $n$ and as $n\sim\ell$, the error term is of order constant for all $M$. Nevertheless, the second series in (7) describes very well and in an explicit form the behavior of $k_{n}$ for large $\ell$ and not very large $n$. We also observe that the leading terms in the first series in (7) are $k_{n}\cong a_{n}-i(4\ell)^{-1}\ln F(a_{n})$ and hence, the real part of the function $\ln F(k)$ determines the sign of the imaginary part of $k_{n}$. As $k$ ranges in $[-r,r]$, the real part of $\ln F(k)$ can change the sign and this is why, our sequence can contain simultaneously eigenvalues and resonances.

4 Examples

Our first example are step-like potentials. Namely, we assume that the potentials $V_{\pm}$ are constant: $V_{-}(x)\equiv-\beta_{-}^{2}$ on $[-1,-0]$ and $V_{+}(x)\equiv-\beta_{+}^{2}$ on $[0,1]$, while outside these segments the functions $V_{\pm}$ vanish identically. Here $\beta_{\pm}\in\mathbb{C}\setminus\{0\}$. It is easy to confirm that condition (3) is equivalent to $\sin\beta_{-}\sin\beta_{+}\neq 0$. The functions $X_{\pm}$ can be found explicitly, which leads to

[TABLE]

Then using the second series in (7), we find a three-terms approximation for $k_{n}$:

[TABLE]

In order to provide an explicit illustration for the sequence of resonances and eigenvalues, we have considered several specific combinations of $\beta_{+}$ and $\beta_{-}$ and computed sequences $k_{n}$ using the iterative procedure (6). The number of iterations was chosen to ensure $|k_{n}-a_{n}-h^{[m]}_{n}(0)|<10^{-16}$. For each combination of $\beta_{+}$ and $\beta_{-}$, values $r$ and $\max|F^{\prime}|$ were estimated by plotting the graphs of $F(k)$ and $F^{\prime}(k)$ in the complex plane. In the examples shown in Fig. 1 the prescribed accuracy was achieved after no more than 15 iterations. The case shown in Fig. 1(a) corresponds to the real-valued, i.e., self-adjoint potential. Respectively, the sequence consists only of resonances and is situated in the lower complex half-plane of wavenumbers $k$. Figure 1(b) illustrates a non-self-adjoint case: here the sequence contains both resonances and eigenvalues.

Our second example is a pair of two delta-interactions, namely, we assume that $V_{\pm}=\beta_{\pm}\delta(x)$, where $\beta_{\pm}$ are complex constants. Condition (3) is ensured as $\beta_{\pm}\neq 0$. Functions $X_{\pm}$ can be found explicitly and $F(k)$ reads

[TABLE]

The first three terms of the second series in (7) are

[TABLE]

Numerical calculation of $k_{n}$ for various $\beta_{\pm}$ produces pictures similar to Fig. 1.

5 Proof of Theorem 1

The functions $X_{-}(0,k)$, $X_{-}^{\prime}(0,k)$, $X_{+}(d_{+},k)$, $X_{+}^{\prime}(d_{+},k)$ are entire in $k$, i.e., they are holomorphic in all $k\in\mathds{C}$. This can be proved by reducing problem (2) for $X_{\pm}$ to the integral Volterra equations

[TABLE]

and studying them in a standard way.

As $k=0$, the denominator in the definition of $F$ is equal to $X_{-}^{\prime}(0,0)X_{+}^{\prime}(d_{+},0)$ and by assumption is nonzero. Hence, the function $F(k)$ is well-defined, holomorphic in the circle $B$ for some $r>0$. We also see that $F(0)=1$. Then we choose $r$ small enough so that the inequality holds:

[TABLE]

By $a_{\pm}(k)$, $b_{\pm}(k)$ we denote the transmission and reflection coefficients for problems (2), which are introduced as the coefficients in the identity

[TABLE]

It is easy to confirm that

[TABLE]

We construct non-trivial solutions to problem (1) as $\psi(x,k)=X_{-}(x+\ell,{k})$, $x<\ell$. Thanks to (2), this function solves the equation in (1) as $x<\ell$ and coincides with $e^{{i}k(x+\ell)}$ as $x<-\ell-d_{-}$. By (12), as $-\ell<x<\ell$, the function $\psi$ reads

[TABLE]

The solution $\psi$ and its derivative $\psi^{\prime}$ are to be continuous at $x=\ell$. Then we apply (12) once again to obtain

[TABLE]

This function solves the equation in (1) as $x>\ell$, is $C^{1}$-smooth at $x=\ell$, and it remains to find its asymptotic behavior as $x\to+\infty$. Again by (12), for $x>\ell+d_{+}$ we get:

[TABLE]

Since the function $\psi$ should not involve $e^{{-}ikx}$ as $x\to+\infty$, the first term in the right hand side of the above formula should vanish. Expressing $a_{\pm}$, $b_{\pm}$ by formulae (13) and dividing the mentioned term by ${b_{-}(k)b_{+}(-k)}$, we arrive at the equation

[TABLE]

The zeroes of this equation are values $k$, for which problem (1) possesses non-trivial solutions.

We proceed to proving (6). Thanks to (11), the function $F(k)$ ranges in a circle of radius $1-e^{-\frac{\pi}{2}}$ centered at $1$ and this circle does not contain the origin. Then the function $h(k):=\ln F(k)$ has no branching points as $k\in\overline{B}$ and is holomorphic. We fix $n$ such that $|n|\leqslant N_{\ell}$. It is clear that $B_{n}\subset B$ and we can rewrite equation (14) as

[TABLE]

Let us prove that the function is a contracting mapping on $\overline{B_{0}}$. It follows from (11) that

[TABLE]

and hence, $|h_{n}(k)|\leqslant\tfrac{\pi}{4\ell}$ as $k\in\overline{B}$. This means that $h_{n}(z)\in\overline{B_{0}}$ as $z\in\overline{B}$. By the Lagrange formula and (11), for all $z_{1},z_{2}\in\overline{B_{0}}$, we obtain:

[TABLE]

where $z_{*}$ is some point in the segment connecting the points $z_{1}$ and $z_{2}$. Hence, by (5), the function $h_{n}(\cdot+a_{n})$ is a contracting mapping. By the contracting mapping principle we obtain immediately that equation (15) possesses the unique solution $z_{n}$ in $\overline{B_{0}}$ and therefore, $k_{n}=z_{n}+a_{n}$ is the unique solution of (14) in $\overline{B_{n}}$. The solution $z_{n}$ can be represented as the limit of $h_{n}^{[m]}(0)$ as $m\to\infty$ and this proves representation for $k_{n}$ in (6). Applying (17) and the estimate $|z_{n}|\leqslant\tfrac{\pi}{4\ell}$ to the identity $z_{n}-h_{n}^{[m]}(0)=h_{n}^{[m]}(z_{n})-h_{n}^{[m]}(0)$, we arrive at the estimate in (6).

Let us prove (7), (8), (9). By the standard estimate for the derivatives of a holomorphic function applied to the function $h$ and the circle $B_{n}$ and by (16) we get:

[TABLE]

and this proves the stated convergence of the first series in (7). By $\tilde{z}_{n}$ we denote the sum of this series; the above estimates also imply that $|\tilde{z}_{n}-a_{n}|\leqslant\frac{\pi}{4\ell}\ln 2<\frac{\pi}{4\ell}$, $\tilde{z}_{n}\in B_{n}$. Let us prove that $\tilde{z}_{n}$ solves equation (15); due to the uniqueness of the root in $\overline{B_{0}}$, this will imply $\tilde{z}_{n}=z_{n}$ and will prove the first identity in (7).

The function $h_{n}$ is holomorphic and $\tilde{z}_{n}$ is holomorphic in $\ell$. Then we can write the Taylor expansion employing the Faà di Bruno formula:

[TABLE]

where $\Pi_{m}$ is the set of all (unordered) partitions of an $m$-element set, the writing ‘$\theta\in\Theta$’ means that $\theta$ runs through the list of all parts in a partition $\Theta$. By $|\Theta|$ and $|\theta|$ we denote the cardinalities of these sets. In view of the Leibnitz rule, we need to prove one of the following equivalent identities

[TABLE]

and it is sufficient to check the latter identity for $h=1$. For each $p\in\mathds{N}$, the quantity $p^{p-1}$ is the number of all (ordered) ways of splitting a $p$-element set into $p$ blocks including possible empty blocks. The set of all such splittings is denoted by $\hat{\Pi}_{k}$ and $p^{p-1}=\sum\limits_{\tau\in\hat{\Pi}_{k}}1$. Then we can rewrite the right hand side in (18) with $h=1$ as

[TABLE]

For each $\Theta=(\theta_{1},\ldots,\theta_{p})\in\Pi_{m}$, the sets $\tau_{1}\in\Pi_{\theta_{1}},\ldots,\tau_{p}\in\Pi_{\theta_{p}}$ provide one of possible splittings of an $(m-p)$-element set into $m$ blocks including possible empty blocks. These splittings are in one-to-one correspondence with the sets $\{\tau_{1},1,\tau_{2},2,\ldots,\tau_{p},p\}$ being splittings of an $m$-element set. The total number of the latter sets is $\tfrac{(m-1)!}{(p-1)!(m-p)!}$, while the total number of ways of splitting an $(m-p)$-element set into $m$ blocks including possible empty blocks is $m^{m-p}$. Hence,

[TABLE]

Substituting this identities into (19), we immediately arrive at (18) and this proves the first identity in (7). For the coefficients in the first series in (7) we have their Taylor expansions:

[TABLE]

Substituting this formulae in the first series in (7) and collecting the coefficients at the like powers of $\ell$, we arrive immediately at the second identity in (7). The series in (7) can be regarded as a Taylor series of $k_{n}$ at some point $a_{n}$ written in powers of $-\tfrac{i}{4\ell}$ and as a similar series at zero. Then estimates (8), (9) are just standard estimates for the remainder in the Lagrange form. The proof of Theorem 1 is complete.

Acknowledgments

We thank a referee for useful remarks allowed us to improve the original version of the paper. The work of D.A.Z. is supported by Russian Foundation for Basic Research, project No. 19-02-00193$\backslash$19.