Spacing gain and absorption in a simple $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetric model: spectral singularities and ladders of eigenvalues and resonances

We consider stationary scattering on a spatially localized $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetric potential $V(x)$:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \displaystyle \label{2.1} -\psi'' + V(x)\psi = \lambda \psi,\qquad \lambda=k^2, \qquad x\in\mathbb{R}, \nonumber \end{align} \tag{ 2.1 }$$

where $\lambda$ is the eigenvalue and k is the complex wavenumber. The complex potential $V(x)$ describes the internal structure of the waveguide. We consider a pair of geometrically identical elements with gain and loss separated by some distance. Since in this work we are primarily interested in the role of the gain-to-loss distance, we assume that the width of each element is scaled to unity. In order to model this situation, we introduce a complex-valued function $W(x)$ compactly supported in the interval $(0, 1)$, and consider a localized complex potential in the form (see figure 1)

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\le}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\leq}{\leqslant} \newcommand{\vs}{\varsigma} \displaystyle \label{2.2} V(x) = \left\{\begin{array}{@{}ll@{}} W(x+\ell+1)\qquad &{{\rm for}}\quad x\in (-\ell-1,-\ell), \nonumber \\ \overline{W}(-x+\ell+1)\qquad &{{\rm for}}\quad x\in (\ell,\ell+1), \nonumber \\ \hphantom{W(x+}0 &{{\rm for~all~other}}~ x, \end{array}\right. \nonumber \end{align} \tag{ 2.2 }$$

where the bar is the complex conjugation, and $\newcommand{\g}{\gamma} \renewcommand{\geq}{\geqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\ge}{\geqslant} \newcommand{\e}{{\rm e}} \ell\geqslant 0$ is the half-distance between the two elements (in order to shorten the discussion, in what follows we will use term 'gain-to-loss distance' for $\newcommand{\e}{{\rm e}} \ell$); the case $\newcommand{\e}{{\rm e}} \ell=0$ corresponds to the adjacent elements. Potential $V(x)$ satisfies the standard condition of $\newcommand{\PT}{\mathcal{PT}} \PT$ symmetry: $V(x)=\overline{V}(-x)$. In figure 1 we show a schematic of the potential consisting of active ('gain') and absorptive ('losses') elements.

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Spectral singularities, finite-width resonances, and eigenvalues are associated with non-trivial solutions $\newcommand{\p}{\partial} \psi(x)$ of equation (2.1) with the following behavior at infinity:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \displaystyle \label{2.3} \psi(x)=C_- {\rm e}^{{\rm i}kx}, \quad {{\rm for~}} x<-\ell-1, \quad {{\rm and\quad~}} \psi(x)=C_+{\rm e}^{-{\rm i}kx}, \quad {{\rm for~}} x>\ell+1, \nonumber \end{align} \tag{ 2.3 }$$

where C₊ and C₋ are constants. Values of k in the upper complex half-plane, Im$\,k>0$, are associated with exponentially growing generalized eigenfunctions, which in the scattering theory correspond to finite-width resonances. The case Im$\,k=0$ corresponds to spectral singularities, i.e. to the zero-width resonances inside the contionuous spectrum, i.e. at $\newcommand{\g}{\gamma} \renewcommand{\geq}{\geqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\ge}{\geqslant} \lambda=k^2\geqslant 0$ [1]. Positive and negative k correspond to the coherent perfect absorber (antilaser) and laser solutions, respectively. Values of k in the lower complex half-plane, Im$\,k<0$, correspond to discrete eigenvalues associated with spatially localized bound states.

Equations (2.1) and (2.2) with condition (2.3) can be solved as

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \displaystyle \psi(x,\ell)={\rm e}^{{\rm i} k (x+\ell+1)}, \qquad x<-\ell-1, \qquad \psi(x,\ell)=\psi_-(x+\ell), \qquad -\ell-1<x<-\ell,\nonumber \\ \psi(x,\ell)=C {\rm e}^{-{\rm i} k (x-\ell-1)}, \qquad x>\ell+1, \qquad \psi(x,\ell)=C\psi_+(x-\ell), \qquad \hphantom{1,.} \ell<x<\ell+1, \nonumber \end{align*}$$

where C is some constant and functions $\newcommand{\p}{\partial} \psi_-$ and $\newcommand{\p}{\partial} \psi_+$ describe the solution inside the active and absorptive cells; their specific shapes are determined by function W and are independent of $\newcommand{\e}{{\rm e}} \ell$. The requirement of the continuity of the wave function $\newcommand{\p}{\partial} \psi(x)$ and of its derivative implies that $\newcommand{\p}{\partial} \psi_-(-1)=1$, $\newcommand{\p}{\partial} \psi_-'(-1)={\rm i}k$, $\newcommand{\p}{\partial} \psi_+(1)=1$, and $\newcommand{\p}{\partial} \psi_+'(1)=-{\rm i}k$.

In the central region, the requirement of continuity of $\newcommand{\p}{\partial} \psi$ and $\newcommand{\p}{\partial} \psi'$ implies that for $\newcommand{\e}{{\rm e}} |x|<\ell$ the solution has the form

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \newcommand{\p}{\partial} \displaystyle \label{2.8} \psi(x)=\psi_-(0)\cos k (x+\ell)+ \psi_-'(0)\frac{\sin k(x+\ell)}{k}. \nonumber \end{align} \tag{ 2.4 }$$

At the same time, the solution for $\newcommand{\e}{{\rm e}} |x|<\ell$ can be also represented as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \newcommand{\p}{\partial} \displaystyle \label{2.9} \psi(x)=C\psi_+(0)\cos k (x-\ell)+ C\psi_+'(0)\frac{\sin k(x-\ell)}{k}. \nonumber \end{align} \tag{ 2.5 }$$

Equations (2.4) and (2.5) are compatible only if k solves the equation

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \displaystyle \label{eq:main} \big(\psi_-'(0)+{\rm i} k \psi_-(0)\big) \big(\psi_+'(0)-{\rm i} k \psi_+(0)\big)= {\rm e}^{-4{\rm i} k\ell} \big(\psi_+'(0)+{\rm i} k \psi_+(0)\big) \big(\psi_-'(0)-{\rm i} k \psi_-(0)\big). \nonumber \end{align} \tag{ 2.6 }$$

The values $\newcommand{\p}{\partial} \psi_-(0)$, $\newcommand{\p}{\partial} \psi'_-(0)$, $\newcommand{\p}{\partial} \psi_+(0)$, $\newcommand{\p}{\partial} \psi'_+(0)$ are independent of $\newcommand{\e}{{\rm e}} \ell$. This means that the gain-to-loss distance $\newcommand{\e}{{\rm e}} \ell$ enters the main equation (2.6) only in the exponential term $\newcommand{\e}{{\rm e}} {\rm e}^{-4{\rm i} k\ell}$, which is periodic in $\newcommand{\e}{{\rm e}} \ell$. This observation which will be used in what follows.

The simplest implementation of potential (2.2), which will be considered in the rest of this paper, corresponds to the case when $W(x)$ is a rectangular function with a purely imaginary amplitude, i.e. $\newcommand{\g}{\gamma} W(x) = {\rm i}\gamma$ for $x\in(0,1)$ and $W(x)=0$ outside this interval. Then $\newcommand{\g}{\gamma} \g>0$ is the gain-and-loss amplitude, which determines the gain strength for the left rectangle and the absorption rate for the right rectangle. The resulting potential $V(x)$ is a generalization of the $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetric step function with adjacent gain and loss regions, which corresponds to $\newcommand{\e}{{\rm e}} \ell=0$ and was considered earlier in [1, 6, 45, 46, 51]. In the present paper, we are concerned with the effect of the nonzero gain-to-losses distance $\newcommand{\e}{{\rm e}} \ell>0$. The obtained generalized $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetric step potential can be of relevance in various physical settings. First, it arises in the study of the scattering of TE electromagnetic waves in a rectangular waveguide containing gain and losses such that the complex-valued dielectric permittivity changes along the direction of the propagation. This can be implemented by filling regions of the waveguide with resonant atoms [1, 6] (the gain and loss media are separated by vacuum in order to create nonzero distance $\newcommand{\e}{{\rm e}} \ell$). Another well-established setting, where $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetric potential (2.2) can arise, corresponds to the propagation of a paraxial optical beam in a system of coupled waveguides with gain and losses [7, 8]. In this case the variation of complex-valued dielectric permittivity takes place in the transverse direction, while the beam propagates in the longitudinal direction. Similar bicentric $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetric potentials (but in the idealized double-delta function form) have been also considered in the theory of Bose–Einstein condensates, where the domains with gain and losses correspond to the pump and absorption of particles, respectively [52, 53].

For the generalized $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetric step potential, functions $\newcommand{\p}{\partial} \psi_-$ and $\newcommand{\p}{\partial} \psi_+$ can be evaluated easily:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \newcommand{\p}{\partial} \newcommand{\g}{\gamma} \displaystyle \psi_-(x)=\cos\sqrt{k^2-{\rm i}\g}(x+1) +{\rm i} k \frac{\sin\sqrt{k^2-{\rm i}\g}(x+1)}{\sqrt{k^2-{\rm i}\g}}, \quad -1<x<0, \nonumber \end{align} \tag{ 2.7 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \newcommand{\p}{\partial} \newcommand{\g}{\gamma} \displaystyle \psi_+(x)= \cos\sqrt{k^2+{\rm i}\g} (x-1) -{\rm i} k \frac{\sin\sqrt{k^2+{\rm i}\g}(x-1)}{\sqrt{k^2+{\rm i}\g}}, \quad \hphantom{.11} 0<x<1. \nonumber \end{align} \tag{ 2.8 }$$

Using these formulae to compute $\newcommand{\p}{\partial} \psi_-(0)$, $\newcommand{\p}{\partial} \psi'_-(0)$, $\newcommand{\p}{\partial} \psi_+(0)$, $\newcommand{\p}{\partial} \psi'_+(0)$ and substituting the latter into equation (2.6), we arrive at a final equation for k:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \newcommand{\p}{\partial} \newcommand{\g}{\gamma} \displaystyle \label{2.10} &F(k,\ell,\g)=0,\qquad F(k,\ell,\g):=F_-(k,\g)F_+(k,\g)-{\rm e}^{-4{\rm i}k\ell} F_0(k,\g), \nonumber \\ & F_\pm(k,\g):=2{\rm i}k\cos\sqrt{k^2\pm {\rm i}\g} -(2k^2\pm {\rm i}\g)\frac{\sin\sqrt{k^2\pm {\rm i}\g}}{\sqrt{k^2\pm {\rm i}\g}}, \quad F_0(k,\g):= \g^2 \frac{\sin\sqrt{k^2-{\rm i}\g}}{\sqrt{k^2-{\rm i}\g}} \, \frac{\sin\sqrt{k^2+{\rm i}\g}}{\sqrt{k^2+{\rm i}\g}}.\nonumber \end{align} \tag{ 2.9 }$$

Our next step is a detailed study of the zeros of function F.

According to remark 2, as $\newcommand{\g}{\gamma} \g=0$, there is a single zero k  =  0. It corresponds to a resonance and the associated solution to equation (2.1) is the constant function. As soon as $\newcommand{\g}{\gamma} \g$ is positive, no matter how small it is, there arise infinitely many zeros of the function F, see lemma 1, and we still have a resonance at k  =  0 but now the associated solution of equation (2.1) is non-constant. Despite the potential $V$ is a regular perturbation as $\newcommand{\g}{\gamma} \g$ is small, it influences the general spectral picture quite essentially producing at once infinitely many zeros. All these zeros are located symmetrically with respect to the imaginary axis and are continuous in $\newcommand{\g}{\gamma} \g$ and $\newcommand{\e}{{\rm e}} \ell$. For the corresponding eigenvalues and resonances given by $\lambda=k^2$, this property means that we deal with complex-conjugate pairs of resonances and eigenvalues, as it is usual for $\mathcal{PT}$-symmetric systems.

By lemma 2, for small $\newcommand{\g}{\gamma} \g$, we have one more pure imaginary zero close to k  =  0. This zero is holomorphic in $\newcommand{\g}{\gamma} \g$ and the leading terms of its Taylor series are given by (3.3). Since $\newcommand{\g}{\gamma} \newcommand{\e}{{\rm e}} (\ell+\frac{1}{3})\g>0$, this zero corresponds to a resonance. The corresponding value $\lambda$ is negative and lies close or next to the bottom of the essential spectrum.

Lemmata 1 and 3 state the following important facts. The zeros can accumulate at infinity only, that is, each circle of a fixed radius in the complex plane contains only finitely many zeros. Except for zeros in the circle $\newcommand{\g}{\gamma} |k|<\sqrt{\g} r$, where r is introduced in (3.6), all other zeros are located in the upper half-plane in the exponential sector

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\IM}{{\rm Im}} \newcommand{\g}{\gamma} \displaystyle \sqrt{\g}{\rm e}^{(\ell+1)\IM\, k}<|k|<\frac{47}{25} \sqrt{\g}{\rm e}^{(\ell+1)\IM\, k},\qquad |k|<\frac{76}{73}\g {\rm e}^{(\ell+\frac{9}{8})|k|}. \nonumber \end{align*}$$

The second inequality deserves a special consideration. Let $\newcommand{\g}{\gamma} \g$ be small enough and consider the zeros outside the circumference $\newcommand{\g}{\gamma} |k|=\sqrt{\g} r$. Then the inequality implies

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\g}{\gamma} \displaystyle \frac{39}{20}<\frac{76}{73}\g^\frac{1}{2} {\rm e}^{(\ell+\frac{9}{8})|k|}\qquad \Rightarrow\qquad |k|>\frac{1}{\ell+\frac{9}{8}}\ln \frac{2847\g^{-\frac{1}{2}}}{1520} \nonumber \end{align*}$$

and hence, except the zeros discussed in lemma 2, all other zeros are located far from k  =  0 at distances of order $\newcommand{\g}{\gamma} O(|\ln\g|)$. This means that for small $\newcommand{\g}{\gamma} \g$, infinitely many zeros emerge from infinity in the upper half-plane and only two of them come from zero and are located in the upper half-plane too.

Since all zeros with $\newcommand{\IM}{{\rm Im}} \renewcommand{\leq}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\le}{\leqslant} \IM\, k \leqslant 0$ are located in the aforementioned circle $\newcommand{\g}{\gamma} |k|<\sqrt{\g} r$, for all $\newcommand{\g}{\gamma} \g>0$ and $\newcommand{\g}{\gamma} \renewcommand{\geq}{\geqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\ge}{\geqslant} \newcommand{\e}{{\rm e}} \ell\geqslant 0$, there are only finitely many such zeros. They correspond to the eigenvalues (bound states), and for them and inequality (3.1) should be satisfied as well. All eigenvalues are complex-valued since according lemma 7, there are no pure imaginary zeroe in the lower complex half-plane. Real zeros correspond to spectral singularities. Thus, for all $\newcommand{\g}{\gamma} \renewcommand{\geq}{\geqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\ge}{\geqslant} \newcommand{\e}{{\rm e}} \ell\geqslant 0$ and $\newcommand{\g}{\gamma} \g>0$ the system has infinitely many resonances and finitely many bound states and spectral singularities. The size of the aforementioned circle, in which the zeros with $\newcommand{\IM}{{\rm Im}} \renewcommand{\leq}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\le}{\leqslant} \IM\, k \leqslant 0$ are located, increases proportionally to $\newcommand{\g}{\gamma} \g$ (see the definition of r in (3.6)). Hence, by increasing $\newcommand{\g}{\gamma} \g$ we have more chances to get the zeros in the lower half-plane and as we shall discuss below, this is indeed the case.

Here we discuss the behavior of the zeros as $\newcommand{\e}{{\rm e}} \ell$ grows. Lemma 6 states that below the line $\newcommand{\IM}{{\rm Im}} \newcommand{\e}{{\rm e}} \IM\, k=-\frac{\ln \frac{\ell\sqrt{C_1}}{\sqrt{C_2}}}{2(1-\ln 2)\ell}$, the function F possesses the same number of zeros as the functions F₋ and F₊ do and these zeros of F converge to the zeros of $\newcommand{\p}{\partial} F_\pm$ as $\newcommand{\e}{{\rm e}} \ell\to+\infty$, see (3.22). As $\newcommand{\e}{{\rm e}} \ell\to+\infty$, the distance from the aforementioned line to the real axis is of order $\newcommand{\e}{{\rm e}} O(\ell^{-1}\ln\ell)$ and is small. A natural question whether there can be some other zeros in the lower half-plane above this line is answered in lemma 4 and the answer is positive. According to this lemma, making $\newcommand{\e}{{\rm e}} \ell$ large enough, we certainly have 2N  +  1 zeros in the vicinity of the real axis, where $\newcommand{\Tht}{\Theta} \newcommand{\g}{\gamma} \newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} N:=\lfloor \frac{1}{2}+\frac{2\Tht(\g)}{\pi}\ell\rfloor$, $\lfloor\cdot\rfloor$ is the integer part of a number. All these zeros are located in the circles $\newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \newcommand{\bi}{\boldsymbol} \big\{k:\, |k-\frac{\pi n}{2\ell}|<\frac{\pi}{4\ell}\big\}$, exactly one zero in each circle, and all these circles are inside a fixed circle $\newcommand{\Tht}{\Theta} \newcommand{\g}{\gamma} \{k:\,|k|<\Tht(\g)\}$. As $\newcommand{\e}{{\rm e}} \ell$ grows, the number of such zeros increases as well but all of them remain in the circle $\newcommand{\Tht}{\Theta} \newcommand{\g}{\gamma} \{k:\,|k|<\Tht(\g)\}$. As $\newcommand{\g}{\gamma} \g$ grows, the size of the latter circle increases as $\newcommand{\g}{\gamma} O(\sqrt{\g})$. For large $\newcommand{\e}{{\rm e}} \ell$, these zeros are approximately given by asymptotic formulae (3.10). As these formulae show, if $\newcommand{\g}{\gamma} \newcommand{\si}{\sigma} \sin\sqrt{2\g}<0$, the zeros are located in the upper half-plane and correspond to resonances. The described behavior is similar to that of Wannier–Stark resonances. Namely, for gain-and-loss amplitude we again have a ladder of resonances accumulating in the vicinity certain zone in the essential spectrum. As $\newcommand{\g}{\gamma} \newcommand{\si}{\sigma} \sin\sqrt{2\g}>0$, these zeros are in the lower half-plane and correspond to eigenvalues. The number of these zeros guaranteed by lemma 4 grows as $\newcommand{\e}{{\rm e}} O(\ell)$ as $\newcommand{\e}{{\rm e}} \ell\to+\infty$, while the distances between the neighbouring zeros are of order $\newcommand{\e}{{\rm e}} O(\ell^{-1})$. This means that we have either resonances or eigenvalues accumulating to the segment $\newcommand{\Tht}{\Theta} \newcommand{\g}{\gamma} [-\Tht(\g),\Tht(\g)]$ on the real axis, see figures 2(a) and (b). It is especially interesting to notice that by means of a continuous change of the gain-and-loss amplitude one can transform a ladder of resonances to a ladder of eigenvalues or vice versa as shown in figure 2(b). Such a transformation is obviously not possible for Wannier–Stark ladders in self-adjoint operators where complex eigenvalues cannot exist. The peculiar regime $\newcommand{\g}{\gamma} \newcommand{\si}{\sigma} \sin\sqrt{2\g}=0$ corresponds to the situation when the leading order of imaginary part of expansion (3.10) vanishes (curve 2 in figure 2(b)). A delicate analysis shows that in this case the imaginary parts of zeros described by lemma 4 are nonzero and amount to $\newcommand{\e}{{\rm e}} O(\ell^{-5})$. Thus, for $\newcommand{\g}{\gamma} \newcommand{\si}{\sigma} \sin\sqrt{2\g}=0$ we have a ladder of coexisting 'nearly spectral singularities' with extremely small by nevertheless nonzero imaginary parts (genuine spectral singularities with exactly zero imaginary parts will be discussed in section 5).

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It is interesting to compare this spectral picture with that in the limiting situation $\newcommand{\e}{{\rm e}} \ell=+\infty$ and the first issue is which an operator are to be regarded as limiting ones. Such limiting operators are to be treated in the norm resolvent sense, that is, we should find an approximation for the resolvent of our operator as $\newcommand{\e}{{\rm e}} \ell\to+\infty$. According the results of [21], as $\newcommand{\e}{{\rm e}} \ell\to+\infty$, the solution of the equation

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \displaystyle -u''+Vu-\lambda u=f\quad{{\rm in}}\quad\mathbb{R},\quad u\in H^2(\mathbb{R}),\quad f(x):=f_-(x+\ell)+f_0(x)+f_+(x-\ell),\quad f_0,\,f_\pm\in L_2(\mathbb{R}), \nonumber \end{align*}$$

is approximated in the norm resolvent sense up to a small error by $\newcommand{\e}{{\rm e}} u_-(x+\ell)+u_0(x)+u_+(x-\ell)$. Here $\newcommand{\p}{\partial} u_0, u_\pm\in H^2(\mathbb{R})$ solve the equations

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \displaystyle -u_0''-\lambda u_0=f_0,\quad -u_\pm''+V_\pm u_\pm -\lambda u_\pm=f_\pm\quad{{\rm in}}\quad \mathbb{R}, \nonumber \end{align*}$$

where the potentials $\newcommand{\p}{\partial} V_\pm$ were introduced in (3.17). Hence, the limiting operator is in fact a direct sum of three operators $\newcommand{\Op}{\mathcal{H}} \Op_-\oplus\Op_0\oplus\Op_+$, $\newcommand{\Op}{\mathcal{H}} \Op_0:=-\frac{{\rm d}^2\ }{{\rm d}x^2}$, $\newcommand{\p}{\partial} \newcommand{\Op}{\mathcal{H}} \Op_\pm:=-\frac{{\rm d}^2\ }{{\rm d}x^2}+V_-$. The essential spectra of all three operators are $[0,+\infty)$. The operator $\newcommand{\Op}{\mathcal{H}} \Op_0$ has the only resonance at k  =  0, while the resonances and the eigenvalues of the operators $\newcommand{\p}{\partial} \newcommand{\Op}{\mathcal{H}} \Op_\pm$ are expressed via the zeros of the functions $\newcommand{\p}{\partial} F_\pm$ by the formula $\lambda=k^2$, see lemma 5. In view of such an approximation, it would be natural to expect that as $\newcommand{\e}{{\rm e}} \ell\to+\infty$, the above discussed zeros $\newcommand{\g}{\gamma} \newcommand{\e}{{\rm e}} k_n(\ell,\g)$ converge to similar real zeros of the functions $\newcommand{\p}{\partial} F_\pm$ or to zero. This is not true! Indeed, by lemma 5, each of the functions $\newcommand{\p}{\partial} F_\pm$ has at most one real zero, while the number of the zeros $\newcommand{\g}{\gamma} \newcommand{\e}{{\rm e}} k_n(\ell,\g)$ in the vicinity of this segment increases as $\newcommand{\e}{{\rm e}} \ell\to+\infty$. Thus, regarding the case $\newcommand{\e}{{\rm e}} \ell=+\infty$ and eigenvalues equations for the operators $\newcommand{\p}{\partial} \newcommand{\Op}{\mathcal{H}} \Op_\pm$ as limiting, for sufficiently large finite $\newcommand{\e}{{\rm e}} \ell$ we get a large number of eigenvalues or resonances emerging from internal points of the zone $\newcommand{\Tht}{\Theta} [0,\Tht^2(g)]$ in essential spectrum. And at most three points (including zero) are spectral singularities for the limiting equations. In other words, here we have eigenvalues emerging from ordinary points in the essential spectrum not being spectral singularities for the unperturbed operators. And while in the case of Wannier–Stark resonances their existence is due to the eigenvalues of the operator $-\frac{{\rm d}^2\ }{{\rm d}x^2}+\varepsilon x$ accumulating on the real line as $\varepsilon\to+0$, this surely not the case in our model.

Here we just make one more important observation. The presence of the mentioned zero-width resonances, or real zeroes of F, could be regarded as an explanation or the reason for the existence of the above described ladder of resonances and eigenvalues. However, this is not the case! Indeed, for $\newcommand{\g}{\gamma} \newcommand{\p}{\partial} \frac{\pi^2}{2} < \gamma< \gamma_*\approx 11.561$, the function F does have a ladder of resonances, but as we shall show in the next section, it has no real zeros. So, our ladder of resonances or eigenvalues exists due to some other reason and the existence of the zero-width resonances is mostly the implication of the presence of such a ladder and its behavior as $\newcommand{\g}{\gamma} \g$ varies.

In this section we study real zeros of the function F corresponding to spectral singularities, i.e. to zero-width resonances. For such zeros, equation (2.9) is a pair of two real equations for one real variable k and two parameters $\newcommand{\e}{{\rm e}} \ell$ and $\newcommand{\g}{\gamma} \g$. Thanks to the symmetry of the zeros with respect to the imaginary axis, it is sufficient to find only positive real resonances since the negative ones are located symmetrically with respect to the origin. Similar to the proof of lemma 5, for real positive k we make change (3.17) and rewrite equation (2.9) in the following form:

$$\begin{align*} \newcommand{\e}{{\rm e}} \renewcommand{\le}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\leq}{\leqslant} \newcommand{\si}{\sigma} \newcommand{\g}{\gamma} \displaystyle (1-2u^{-4})\cos\beta u^{-1}&+(2u^4-1)\cosh\beta u + 2{\rm i} \sqrt{1-u^4}\left(u^2\sinh\beta u - u^{-4}\sin\beta u^{-1}\right) \nonumber \\ &-{\rm e}^{-2{\rm i}\beta\ell\sqrt{u^{-2}-u^2}} \left(\cosh(\beta u)-\cos\beta u^{-1}\right)=0,\qquad \beta=\sqrt{2\g}. \nonumber \end{align*}$$

Taking the real and imaginary part of this equation and multiplying the equation by u⁴, we obtain:

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \renewcommand{\le}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\leq}{\leqslant} \newcommand{\si}{\sigma} \displaystyle \label{4.3} &(u^4-2)\cos\beta u^{-1}+u^4(2u^4-1)\cosh\beta u =u^4\cos\left(2\beta\ell\sqrt{u^{-2}-u^2}\right)\left(\cosh \beta u-\cos\beta u^{-1}\right), \nonumber \\ &2\sqrt{1-u^4}\Big(u^6\sinh\beta u-\sin\beta u^{-1}\big) =-u^4\sin\left(2\beta\ell\sqrt{u^{-2}-u^2}\right)\left(\cosh \beta u-\cos\beta u^{-1}\right). \nonumber \end{align} \tag{ 5.1 }$$

This is a system of two real equations with three real variables. If we are given $\newcommand{\e}{{\rm e}} (\beta,\ell)$ and we try to find u, the system is overdetermined and does not necessary have a root. In other words, it is solvable with respect to u only if $\newcommand{\e}{{\rm e}} (\beta,\ell)$ are located on some (solvability) curves. In order to avoid working with an overdetermined system, in what follows we regard (5.1) as a system for two unknown variables with one parameter.

To find the curves in $\newcommand{\e}{{\rm e}} (\beta,\ell)$ plane, on which equation (5.1) are solvable with respect to u, we shall regard u as a parameter and $\newcommand{\e}{{\rm e}} (\beta,\ell)$ as unknown variables. We take the sum of squares of equation (5.1) and divide the result by (1  −  u⁴). We also divide equation (5.1). This leads us to a pair of equations:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \displaystyle \label{4.4} &u^4(1-u^4) \cosh\beta u\cos\beta u^{-1}-2u^6 \sinh\beta u\sin\beta u^{-1} +1-u^{12}=0,\nonumber \\ &\frac{2\sqrt{1-u^4}(u^6\sinh\beta u-\sin\beta u^{-1})}{(2-u^4)\cos\beta u^{-1}+u^4(1-2u^4)\cosh\beta u}=\tan \Big(2\beta\ell\sqrt{u^{-2}-u^2}\Big).\nonumber \end{align} \tag{ 5.2 }$$

The second equation can be solved explicitly with respect to $\newcommand{\e}{{\rm e}} \ell$:

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\le}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\leq}{\leqslant} \newcommand{\si}{\sigma} \newcommand{\p}{\partial} \displaystyle \label{4.10} \ell=\frac{1}{2\beta\sqrt{u^{-2}-u^2}} \left(\arctan \frac{2\sqrt{1-u^4}(u^6\sinh\beta u-\sin\beta u^{-1})}{(2-u^4)\cos\beta u^{-1}+u^4(1-2u^4)\cosh\beta u}+\pi n\right), \nonumber \end{align} \tag{ 5.3 }$$

where $n\in\mathbb{N}$ is an arbitrary natural number. As the next lemma states, to make equations (5.2) and (5.3) equivalent to (5.1), we should also assume that

$$\begin{align} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \newcommand{\sign}{{\rm sign}} \newcommand{\si}{\sigma} \displaystyle \label{4.5} (-1)^n=\sign \big((u^4-2)\cos\beta u^{-1}+u^4(2u^4-1)\cosh\beta u\big). \nonumber \end{align} \tag{ 5.4 }$$

Lemma 8. Equations (5.1) are equivalent to (5.2)–(5.4).

Proof. We shortly rewrite equation (5.1) as $A_1=B\cos\alpha$, $\newcommand{\si}{\sigma} A_2=-B\sin\alpha$, where A₁, A₂ are the left-hand sides in (5.1), $\newcommand{\e}{{\rm e}} \alpha=2\beta\ell\sqrt{u^{-2}-u^2}$ and $B=u^4(\cosh\beta u-\cos\beta u^{-1})$. Then equations (5.2) and (5.3) become $A_1^2+A_2^2=B^2$, $\newcommand{\p}{\partial} \alpha=-\arctan\frac{A_2}{A_1}+\pi n$. We have:

$$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle B\cos \alpha=(-1)^n\cos\arctan\frac{A_2}{A_1}=\frac{(-1)^n}{\sqrt{1+\frac{A_2^2}{A_1^2}}} = \frac{(-1)^n|A_1|}{\sqrt{A_1^2+A_2^2}} \nonumber \end{align*}$$

and we get the first equation $A_1=B\cos\alpha$ provided condition (5.4) is satisfied. In the same way we check that the latter condition also ensures the second equation $\newcommand{\si}{\sigma} A_2=-B\sin\alpha$. □

Equation (5.2) is transcendental, and we cannot solve it analytically. Nevertheless, for each $u\in(0,1)$, this is an equation only for a single variable $\beta$, not for two as equation (5.1). So, we propose the following algorithm of recovering the aforementioned solvability curves: choose $u\in(0,1)$, then solve equation (5.2) and recover the sequence of distances $\newcommand{\e}{{\rm e}} \ell$ by formula (5.3) with different integer n. Then the gain-and-loss amplitude $\newcommand{\g}{\gamma} \gamma$ and the corresponding wavenumber k can be readily recovered from $\beta$ and u. In a similar way, one can first fix some value of $\beta$ (i.e. fix the gain-and-loss strength) and then solve equation (5.2) with respect to u and recover $\newcommand{\e}{{\rm e}} \ell$ by (5.3). Equation (5.2) is well-behaved, and for each $\beta$ all its zeros u can be easily found numerically.

Alternatively, as explained below in section 5.5, the values corresponding to spectral singularities can be found systematically by means of the numerical continuation from the limit $\newcommand{\e}{{\rm e}} \ell=0$. However, in this case equation (5.2) is still useful because it allows one to check that all spectral singularities have been found for the given value of the gain-and-loss $\newcommand{\g}{\gamma} \gamma$.

For u  =  0 and u  =  1, the left-hand-side of equation (5.2) is equal respectively to 1 and $\newcommand{\si}{\sigma} -2\sinh\beta\sin\beta$. Then a sufficient condition for the existence of a spectral singularity at the given gain-and-loss amplitude $\newcommand{\g}{\gamma} \gamma$ is $\newcommand{\g}{\gamma} \newcommand{\si}{\sigma} \sin\beta=\sin\sqrt{2\gamma}>0$. At the same time, it is also possible to establish sufficient conditions that forbid the existence of spectral singularities in a certain interval of parameters. In this subsection we prove the existence of two 'forbidden gaps' for the roots of equation (5.2). The first one exists for all $\newcommand{\g}{\gamma} \renewcommand{\geq}{\geqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\ge}{\geqslant} \beta\geqslant 0$ and it states that there is no roots in certain interval. The second gap is a certain interval of values of $\beta$, for which equation (5.2) has no zeros at all.

For the convenience, by $g(u,\beta)$ we denote the left-hand side of equation (5.2). The first 'forbidden gap' is described in the following lemma.

Lemma 9. For all $\newcommand{\g}{\gamma} \renewcommand{\geq}{\geqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\ge}{\geqslant} \beta\geqslant 0$, equation (5.2) has no roots in the interval $\newcommand{\bi}{\boldsymbol} \big[0,(1+\frac{\beta}{4}){}^{-1}\big)$.

Proof. Employing a standard inequality $\newcommand{\si}{\sigma} \renewcommand{\leq}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\le}{\leqslant} a\cos\alpha+b\sin\alpha\leqslant \sqrt{a^2+b^2}$, we estimate the first two terms in equation (5.2) as

$$\begin{align*} \newcommand{\e}{{\rm e}} \renewcommand{\le}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\leq}{\leqslant} \newcommand{\si}{\sigma} \displaystyle u^4(1-u^4) \cosh\beta u\cos\beta u^{-1}-2u^6 \sinh\beta u\sin\beta u^{-1} \leqslant \sqrt{u^8(1-u^4)^2\cosh^2 \beta u+4u^{12}\sinh^2\beta u}. \nonumber \end{align*}$$

Hence, equation (5.2) surely has no roots for values of u satisfying

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \displaystyle \sqrt{u^8(1-u^4)^2\cosh^2 \beta u+4u^{12}\sinh^2\beta u} <1-u^{12}. \nonumber \end{align*}$$

Expressing $\cosh^2 \beta u$ via $\newcommand{\si}{\sigma} \sinh^2 \beta u$ and simplifying this inequality, we obtain $u^4(1+u^4)\cosh \beta u <1+u^{12}$ and hence,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \displaystyle \label{3.37} \cosh \beta u-1 <\frac{1-2u^4+u^8}{u^4},\qquad \sqrt{2}\sinh \beta u <u^{-2}-u^2. \nonumber \end{align} \tag{ 5.5 }$$

Thanks to the estimate $\newcommand{\si}{\sigma} \renewcommand{\leq}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\le}{\leqslant} \sqrt{2}\sinh 2\ln u^{-1}\leqslant u^{-2}-u^2$, $u\in(0,1]$, the last inequality in (5.5) holds once $\frac{\beta}{4}<u^{-1}\ln u^{-1}$. It is easy to confirm that this inequality is true once $u^{-1}>1+\frac{\beta}{4}$. The proof is complete. □

The next lemma is auxiliary and will be employed in studying the second forbidden zone.

Lemma 10. The function $\newcommand{\p}{\partial} g(u,\pi)$ is positive on $[0,1)$.

Proof. We have $\newcommand{\p}{\partial} g(0,\pi)=1$ and by lemma 9, it is positive for $\newcommand{\p}{\partial} u<(1+\frac{\pi}{4}){}^{-1}$. This is why in what follows we consider only the values $\newcommand{\g}{\gamma} \newcommand{\p}{\partial} \renewcommand{\geq}{\geqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\ge}{\geqslant} u\geqslant (1+\frac{\pi}{4}){}^{-1}$. For such values of u we have $\newcommand{\p}{\partial} \renewcommand{\leq}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\le}{\leqslant} \pi\leqslant \pi u^{-1}\leqslant \pi (1+\frac{\pi}{4})<1.79\pi$. As $\newcommand{\p}{\partial} \renewcommand{\leq}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\le}{\leqslant} \frac{3\pi}{2} \leqslant \pi u^{-1}\leqslant \pi (1+\frac{\pi}{4})$, the function $\newcommand{\si}{\sigma} \sin\beta u^{-1}$ is negative, while $\cos \beta u^{-1}$ is positive. Hence, for such values of u, the function $\newcommand{\p}{\partial} g(u,\pi)$ is positive. It remains to consider the values $\renewcommand{\leq}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\le}{\leqslant} \frac{2}{3}<u\leqslant 1$.

For such values of u we first observe the following simple estimates:

$$\begin{align*} \newcommand{\e}{{\rm e}} \renewcommand{\ge}{\geqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\geq}{\geqslant} \newcommand{\si}{\sigma} \newcommand{\p}{\partial} \newcommand{\g}{\gamma} \displaystyle \frac{1-u^4}{1-u}\cos\pi u^{-1}\geqslant -4,\qquad -\frac{\sin \pi u^{-1}}{1-u}=\frac{\sin \pi (u^{-1}-1)}{1-u}\geqslant \pi u. \nonumber \end{align*}$$

Then the function g satisfies the estimate:

$$\begin{align} \newcommand{\e}{{\rm e}} \renewcommand{\ge}{\geqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\geq}{\geqslant} \newcommand{\si}{\sigma} \newcommand{\p}{\partial} \newcommand{\g}{\gamma} \displaystyle \label{3.12} \frac{g(u,\pi)}{u^4(1-u)}\geqslant g_1(u)+g_2(u),\qquad g_1(u):=-4\cosh\pi u +2\pi u^3 \sinh \pi u,\qquad g_2(u):=\frac{1-u^{12}}{u^4(1-u)}. \nonumber \end{align} \tag{ 5.6 }$$

The function g₁(u) is monotonically increasing in $u\in[\frac{2}{3},1]$ since

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \newcommand{\p}{\partial} \displaystyle g_1'(u)=2\pi^2 u^3\cosh\pi u+(6\pi u^2-4\pi)\sinh\pi u>2\pi(\pi u^3+6u^2-2)\sinh\pi u>2\pi\sinh\pi u>0. \nonumber \end{align*}$$

Hence, $\newcommand{\g}{\gamma} \renewcommand{\geq}{\geqslant} \renewcommand{\ge}{\geqslant} \renewcommand{\leq}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\le}{\leqslant} g_1(u)\geqslant g_1\left(\frac{2}{3}\right)>-9.05$. For the function g₂ we have the following representation and estimate:

$$\begin{align*} \newcommand{\e}{{\rm e}} \renewcommand{\le}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\leq}{\leqslant} \renewcommand{\ge}{\geqslant} \renewcommand{\geq}{\geqslant} \newcommand{\g}{\gamma} \displaystyle g_2(u)=1+\sum\limits_{j=1}^{4}(u^{\,j}+u^{-j})+\sum\limits_{j=5}^{8} u^{\,j}\geqslant 9 +\sum\limits_{j=5}^{8}\left(\frac{2}{3}\right)^{\,j}>9.31. \nonumber \end{align*}$$

Two last estimates and (5.6) imply the positivity of the function g for $u\in[\frac{2}{3},1)$. □

Denote

$$\begin{align*} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \displaystyle g_*(u,\beta):=&\beta u^3(1-3u^4)\cosh\beta u \sin \beta u^{-1}+\beta u^5 (3-u^4)\sinh\beta u \cos \beta u^{-1} \nonumber \\ &-2 u^4(1+u^4) \cosh\beta u \cos \beta u^{-1} -6(1+u^{12}). \nonumber \end{align*}$$

The next lemma states the existence of the second forbidden zone.

Lemma 11. Equation (5.2) has no roots as $\newcommand{\p}{\partial} \pi<\beta<\beta_*<5$, where $(u_*, \beta_*)$ is the root of the system of the equations

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \displaystyle \label{3.13} g(u,\beta)=0,\qquad g_*(u,\beta)=0,\qquad u\in[0,1],\qquad \pi<\beta<5, \nonumber \end{align} \tag{ 5.7 }$$

with minimal possible $\beta$. Their approximate values are

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{3.14} \beta_*=4.808\,438,\qquad u_*=0.611\,772. \nonumber \end{align} \tag{ 5.8 }$$

Proof. The function $\newcommand{\p}{\partial} g(u,\pi)$ is positive on $[0,1)$ and $\newcommand{\p}{\partial} g(1,\pi)=0$, see figure 3(a). As $\newcommand{\p}{\partial} \pi<\beta<2\pi$, we have $g(0,\beta)=1>0$ and $\newcommand{\si}{\sigma} g(1,\beta)=-2\sin\beta\sinh\beta>0$. Hence, for $\beta$ close enough to $\newcommand{\p}{\partial} \pi$, the function $g(u,\beta)$ is positive for all $u\in[0,1]$. At the same time, we have $g(0.65,5)<-0.617<0$ and therefore, for $\beta=5$, equation (5.2) possesses at least two roots, one in $(0,0.65)$ and another in $(0.65,1)$. See figure 3(c). We also observe that the function g is jointly continuous in $(u,\beta)$. The above facts means that as $\beta$ grows from $\newcommand{\p}{\partial} \pi$ to 5, at some value $\beta=\beta_*$, the graph of the function g is still located in the upper half-plane but touches the u-axis at some point u  =  u_*, see figure 3(b). The function $g(u,\beta)$ is positive as $\newcommand{\p}{\partial} \pi<\beta<\beta_*$ and $u\in[0,1]$. Then the point u  =  u_* is obviously the global minimum of g, and hence $(u_*,\beta_*)$ is a solution to the system of equations $g(u,\beta)=0$, $\newcommand{\p}{\partial} \frac{\p g}{\p u}(u,\beta)=0$. It is easy to check that $\newcommand{\p}{\partial} g_*=\frac{\p g}{\p u}-6 g$, and hence $(u_*,\beta_*)$ solves system (5.7). These roots can be found numerically and this gives (5.8). The proof is complete. □

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Returning from the auxiliary variable $\beta$ to the gain-and-loss amplitude $\newcommand{\g}{\gamma} \gamma=\beta^2/2$, from lemma 11 we deduce the following important result:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\p}{\partial} \newcommand{\g}{\gamma} \displaystyle \label{eq:gap} {\rm there~ is~ no~ spectral ~singularities~ for\quad } \frac{\pi^2}{2} < \gamma< \gamma_*\approx 11.561. \nonumber \end{align} \tag{ 5.9 }$$

For $\newcommand{\e}{{\rm e}} \ell=0$ spectral singularity can only be obtained for some isolated values of the wavenumber k and the gain-and-loss amplitude $\newcommand{\g}{\gamma} \gamma$ [1]. Several lowest values of $\newcommand{\g}{\gamma} \gamma$ corresponding to the spectral singularities and the associated wavenumbers k are listed in table 1. An important advantage of the more general system with nonzero gain-to-loss distance $\newcommand{\e}{{\rm e}} \ell>0$ consists in the possibility to create a spectral singularity at any wavenumber k given beforehand. Indeed, let us return back to equations (5.2), (5.3) and discuss the following issue: given a point k on the real axis, how do we choose $\beta$ and $\newcommand{\e}{{\rm e}} \ell$ to have a resonance at this point? Equations (5.2) and (5.3) allow us to easily answer this question.

Table 1. Approximate values of gain-and-loss amplitudes $\newcommand{\g}{\gamma} \gamma=\gamma_\star^{(n)}$ and wavenumbers $k=k_\star^{(n)}$, $n=0, 1, \ldots$, corresponding to spectral singularities with lowest $\newcommand{\g}{\gamma} \gamma$ in the limit $\newcommand{\e}{{\rm e}} \ell=0$, see table I in [1].

n	0	1	2
$\newcommand{\g}{\gamma} \gamma_\star^{(n)}$	2.071	13.307	27.783
$k_\star^{(n)}$	1.065	4.318	7.529

We fix k  >  0 and we find the associated value of u by resolving (3.17):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{4.12} u^{-2}-u^2=4k^2\beta^{-2},\qquad u=\beta R^{-1}, \quad {{\rm where~}} R = \sqrt{2k^2+\sqrt{4k^4+\beta^4}}. \nonumber \end{align} \tag{ 5.10 }$$

We divide equation (5.2) by u⁶ and substitute then the above formulae and

$$\begin{align*} \newcommand{\bi}{\boldsymbol} \newcommand{\e}{{\rm e}} \displaystyle \frac{1-u^{12}}{u^6}=\frac{1-u^4}{u^2}\frac{1+u^4+u^8}{u^4}=(u^{-2}-u^2)\big((u^{-2}-u^2)^2+3 \big). \nonumber \end{align*}$$

This gives the equation:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\si}{\sigma} \displaystyle \label{4.11} 2\beta^4 k^2 \cosh(\beta^2R^{-1}) \cos R - \beta^6 \sinh(\beta^2R^{-1}) \sin R +2k^2(16k^4+3\beta^4)=0. \nonumber \end{align} \tag{ 5.11 }$$

An algorithm for creating a resonance at a prescribed point k is as follows. Given k  >  0, we first solve equation (5.11) with respect to $\beta$ and we also find u by (5.10). Then needed values of $\newcommand{\e}{{\rm e}} \ell$ are determined by (5.3) and (5.4).

In order to illustrate this algorithm, we consider a finite interval of wavenumbers $k\in (0, k_1]$, where we set k₁  =  10 for the numerics reported on in what follows. We scan the chosen interval with a sufficiently small step ($\newcommand{\D}{\Delta} \Delta k=0.01$) and for each value of k solve equation (5.11) numerically using the simple dichotomy method. While for each k equation (5.11) might have several roots $\beta$, in our numerical procedure we always choose the minimal positive root, i.e. the one which allows us to achieve the spectral singularity with given k at the smallest possible value of the gain-and-loss amplitude $\newcommand{\g}{\gamma} \gamma=\gamma_{\min}$. Next, we choose the minimal positive distance $\newcommand{\e}{{\rm e}} \ell_{\min}$ which satisfies the conditions (5.3), (5.4) and then we use the periodicity in $\newcommand{\e}{{\rm e}} \ell$ to generate the sequence of larger gain-to-loss distances $\newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \ell_n = \ell_{\min} + n\pi/(2k)$, $n=1,2\ldots$ (see (3.2)). The resulting dependencies $\newcommand{\g}{\gamma} \gamma_{\min}(k)$ and $\newcommand{\e}{{\rm e}} \ell_{\min}(k)$, $\newcommand{\e}{{\rm e}} \ell_n(k)$ are shown in figure 4. The minimal gain-and-loss amplitude $\newcommand{\g}{\gamma} \g_{\min}(k)$ and the minimal gain-to-loss distance $\newcommand{\e}{{\rm e}} \ell_{\min}$ are discontinuous, which means that the small variation in the wavenumuber k might require a significant change either in $\newcommand{\g}{\gamma} \gamma$ or in $\newcommand{\e}{{\rm e}} \ell$. It is especially important that the values of the distance $\newcommand{\e}{{\rm e}} \ell$ are generically different from zero, which points out explicitly that the new degree of freedom offered by the nonzero gain-to-loss distance is important for achieving a spectral singularity at the given wavenumber k.

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A particularly important characteristic of any $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetric structure is the $\newcommand{\PT}{\mathcal{PT}} \PT{\mbox -}$symmetry breaking threshold, i.e. the amplitude of the gain-and-loss corresponding to the 'phase transition' from the purely real to complex spectrum. The best studied scenario of the phase transition is the collision of two real discrete eigenvalues at an exceptional point with the subsequent splitting in a complex-conjugate pair. However, in systems with a nonempty continuous spectrum, the phase transition can also occur through the splitting of a self-dual spectral singularity, which results in a bifurcation of a complex-conjugate pair from an interior point of the continuum [44, 45, 47]. At the moment corresponding to the formation of the spectral singularity, the system operates in the CPA-laser regime [2]. Thus, in such a system, the $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetry breaking threshold at the same time corresponds to the CPA-laser threshold.

Lemma 3 guarantees that the spectrum of our system is real for sufficiently small gain-and-loss amplitudes $\newcommand{\g}{\gamma} \gamma$. Additionally, according to lemma 7, the spectrum does not have any real discrete eigenvalue. Hence, the $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetry breaking is expected to occur through the emergence of a self-dual spectral singularity. In order to identify the $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetry breaking threshold in our system, we start from the limit $\newcommand{\e}{{\rm e}} \ell=0$, where the phase transition takes place at $\newcommand{\g}{\gamma} \gamma_\star^{(0)} \approx 2.072$, see [1, 45] and table 1. Thus, the spectrum with $\newcommand{\e}{{\rm e}} \ell=0$ is purely real and continuous for $\newcommand{\g}{\gamma} \gamma \in [0, \gamma_\star]$, while the increase of the gain-and-loss just above $\newcommand{\g}{\gamma} \gamma_\star^{(0)}$ leads to the bifurcation of a complex-conjugate pair from an interior point of the continuum. The spectral singularity forming at $\newcommand{\g}{\gamma} \gamma_\star^{(0)}$ takes place at wavenumber $k=k_\star^{(0)} \approx 1.065$. Respectively, the complex-conjugate pair of eigenvalues bifurcates from $\lambda_0 = [k_\star^{(0)}]^2$. (Notice that the further increase of $\newcommand{\g}{\gamma} \gamma$ above the next threshold values listed in table 1 leads to the formation of new spectral singularities and, respectively, to bifurcations of new complex-conjugate pairs in the spectrum.)

Next, we use the numerical continuation in $\newcommand{\e}{{\rm e}} \ell$ in order to continue the known solution at $\newcommand{\e}{{\rm e}} \ell=0$ to the domain $\newcommand{\e}{{\rm e}} \ell>0$. The obtained dependence of the threshold value of the gain-and-loss amplitude on distance $\newcommand{\e}{{\rm e}} \ell$ is shown in figure 5(a), where we observe that the phase transition threshold decreases monotonically with the growth of $\newcommand{\e}{{\rm e}} \ell$. This means that introducing an additional space between the gain and loss, one can decrease the $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetry breaking threshold, i.e. achieve the laser-antilaser operation at lower gain-and-loss amplitudes than in a waveguide with the adjacent gain and loss.

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Let us now turn to the description of the general picture of spectral singularities. In order to systematically construct different solutions, we again start from the limit $\newcommand{\e}{{\rm e}} \ell=0$, where the values of $\newcommand{\g}{\gamma} \gamma$ and k corresponding to spectral singularities are known, see table 1. Then we use the periodicity of function $\newcommand{\e}{{\rm e}} {\rm e}^{-4{\rm i}k\ell}$ in $\newcommand{\e}{{\rm e}} \ell$ in order to construct new branches of solutions having no counterparts in the limit $\newcommand{\e}{{\rm e}} \ell=0$, see equation (3.2). This procedure results in a fairly complicated picture containing a multitude of spectral singularities, some part of which (corresponding to relatively small values of the gain-and-loss) is shown in figures 6(a) and (b) as the curves on the plane $\newcommand{\g}{\gamma} \gamma$ versus $\newcommand{\e}{{\rm e}} \ell$ and k versus $\newcommand{\e}{{\rm e}} \ell$.

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Let us describe the structure of the found solutions using the diagram $\newcommand{\g}{\gamma} \gamma$ versus $\newcommand{\e}{{\rm e}} \ell$ in figure 6(a). The multitude of curves shown in this plot can be divided into three groups (plotted with red, blue and green curves) which have been obtained by means of the continuation from three different solutions in the limit $\newcommand{\e}{{\rm e}} \ell=0$. There is a well-visible vertical gap between red and blue curves, which corresponds to the 'forbidden' values of the gain-and-loss amplitudes $\newcommand{\g}{\gamma} \gamma$ found above in (5.9). At the same time, there is no gap between blue and green curves, which results in a multitude of intersections between these curves. The first group of spectral singularities, corresponding to red curves in figure 6(a), is obtained through the continuation from the spectral singularity in the limit $\newcommand{\e}{{\rm e}} \ell=0$ with the smallest gain-and-loss amplitude, i.e. from values $\newcommand{\g}{\gamma} \gamma_\star^{(0)}$ and $k_\star^{(0)}$ in table 1. The leftmost (bold) curve in this group which originates in the limit $\newcommand{\e}{{\rm e}} \ell=0$ is the $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetry breaking threshold which was already shown in figure 5(a). For values of $\newcommand{\g}{\gamma} \gamma$ above this curve, there is always one or more (but finitely many, see section 4) complex-conjugate pairs of eigenvalues in the spectrum. Several curves situated to the right from the bold red curve are obtained using the fact that if $\newcommand{\g}{\gamma} \gamma$ and k are solutions in the limit $\newcommand{\e}{{\rm e}} \ell=0$, then the same values of $\newcommand{\g}{\gamma} \gamma$ and k also correspond to a spectral singularity with $\newcommand{\p}{\partial} \newcommand{\e}{{\rm e}} \ell_n = (n\pi)/(2k)$, $n=1,2,\ldots$. Once a single solution with a new distance $\newcommand{\e}{{\rm e}} \ell_n$ is obtained, a new branch of solutions can be constructed using numerical continuation in $\newcommand{\g}{\gamma} \gamma$ or in $\newcommand{\e}{{\rm e}} \ell$. In the limit $\newcommand{\e}{{\rm e}} \ell \to \infty$ all the red curves in figure 6(a) approach the asymptotic value $\newcommand{\g}{\gamma} \gamma=0$ or at $\newcommand{\g}{\gamma} \newcommand{\p}{\partial} \gamma_*= \pi^2/2\approx 4.935$. Notice that, except for the bold line demarcating the $\newcommand{\PT}{\mathcal{PT}} \PT$-symmetry breaking threshold, none of the red curves can be continued to the limit $\newcommand{\e}{{\rm e}} \ell\to 0$.

The multitude of red curves in figure 6(a) demonstrates explicitly how new spectral singularities emerge with the increase of $\newcommand{\e}{{\rm e}} \ell$. Indeed, drawing an imaginary horizontal line, say, at $\newcommand{\g}{\gamma} \gamma=3$ (see the vertical axis tick in figure 6(a)), we observe that with the increase of $\newcommand{\e}{{\rm e}} \ell$ this line intersects more and more red curves. Each intersection corresponds to the values of $\newcommand{\g}{\gamma} \gamma$ and $\newcommand{\e}{{\rm e}} \ell$ at which some root k crosses the real axis and goes down from the upper to lower complex half-plane. Thus, a finite-width resonance transforms to a complex eigenvalue through the spectral singularity (we recall again that in view of $\newcommand{\PT}{\mathcal{PT}} \PT$ symmetry any root $k\ne 0$ crosses the real line simultaneously with its counterpart $-\bar{k}$, i.e. the corresponding spectral singularity is self-dual). In order to illustrate this process, in figure 7 we show the evolution of three numerically found complex zeros of function $\newcommand{\g}{\gamma} \newcommand{\e}{{\rm e}} F(k, \gamma, \ell)$ under the increase of $\newcommand{\e}{{\rm e}} \ell$. In this figure the imaginary part of each complex zero changes sign from positive to negative and then asymptotically approaches zero (remaining negative). In the limit of large $\newcommand{\e}{{\rm e}} \ell$, this behavior agrees with expansion (3.10) of lemma 4, where, for the chosen value of $\newcommand{\g}{\gamma} \gamma$, we have $\newcommand{\g}{\gamma} \newcommand{\si}{\sigma} \sin\sqrt{2\gamma}>0$. Thus the growing distance $\newcommand{\e}{{\rm e}} \ell$ results in a sequence of self-dual spectral singularities and in the increasing number of complex-conjugate eigenvalues in the spectrum.

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The second group of spectral singularities (blue curves in figure 6(a)) was obtained by means of the continuation from the next solution in the limit $\newcommand{\e}{{\rm e}} \ell=0$, i.e. from $\newcommand{\g}{\gamma} \gamma_\star^{(1)}$ and $k_\star^{(1)}$ in table 1. Again, one of the curves (the leftmost bold curve in figure 6(a)) was obtained through the direct continuation from the limit $\newcommand{\e}{{\rm e}} \ell=0$, while other blue curves were generated using the periodicity of function $\newcommand{\g}{\gamma} \newcommand{\e}{{\rm e}} F(k, \gamma, \ell)$ in $\newcommand{\e}{{\rm e}} \ell$ and cannot be continued to the limit $\newcommand{\e}{{\rm e}} \ell\to0$. In the limit $\newcommand{\e}{{\rm e}} \ell\to\infty$ the blue curves approach the horizontal asymptotes $\newcommand{\g}{\gamma} \newcommand{\p}{\partial} \gamma=2\pi^2\approx19.739$ and $\newcommand{\g}{\gamma} \newcommand{\p}{\partial} \gamma=9\pi^2/2\approx 44.413$. In the $\newcommand{\g}{\gamma} \newcommand{\e}{{\rm e}} (\gamma, \ell)$-plane the gain-and-loss amplitudes corresponding to the group of blue curves are well separated from those for the red curves: indeed, all red curves are bounded from above by the asymptote $\newcommand{\g}{\gamma} \newcommand{\p}{\partial} \gamma= \pi^2/2\approx 4.935$, while all blue curves are bounded from below by $\newcommand{\g}{\gamma} \gamma_*\approx 11.561$, see (5.9). Thus, the emergence of new spectral singularities with the increase of $\newcommand{\e}{{\rm e}} \ell$ is sensitive to the value of the gain-and-loss amplitude and does not occur for the gain-and-loss amplitudes lying in the gap between the red and blue curves.

In comparison with the red curves discussed above, the curves from the blue group in figure 6(a) feature more complicated behavior and, in particular, can intersect each other (and also intersect the curves from the next, third group of green curves discussed below). The intersections between the blue curves occur for the gain-and-loss amplitudes in the interval $\newcommand{\g}{\gamma} \newcommand{\p}{\partial} \renewcommand{\leq}{\leqslant} \newcommand{\re}{{\rm Re}} \renewcommand{\le}{\leqslant} 11.561 \lessapprox \gamma < 9\pi^2/2\approx 19.739$, where $\newcommand{\g}{\gamma} \newcommand{\si}{\sigma} \sin\sqrt{2\gamma}$ is negative. At first glance, this might seem to contradict to the expansion (3.10) of lemma 4, which suggests that in this case the multitude of complex zeros accumulate in the upper complex half-plane with the growth of $\newcommand{\e}{{\rm e}} \ell$. However, this apparent contradiction is resolved if we trace the behavior of the complex roots more closely. Indeed, choosing for an example $\newcommand{\g}{\gamma} \gamma =16$ (see the vertical axis tick in figure 6(a)) and computing several complex roots under the increase of $\newcommand{\e}{{\rm e}} \ell$, we observe that each considered root first goes down from the upper half-plane to the lower one but then again returns to the upper half-plane, see figure 8(b) and (b₁). Thus, in this interval of the gain-and-loss amplitudes, the increase of $\newcommand{\e}{{\rm e}} \ell$ results either in the transformation from the resonance to the eigenvalue or to the opposite process, i.e. to the disappearance of the complex-conjugate pair. Respectively, the intersection between the two blue curves corresponds to two coexisting spectral singularities, i.e. to the moment when one complex-conjugate pair of eigenvalues disappears and another pair (with different k) emerges. For sufficiently large $\newcommand{\e}{{\rm e}} \ell$ the imaginary part of each considered root remains positive and approaches zero, in accordance with expansion (3.10). Thus, in this interval of the gain-and-loss strengths, the limit $\newcommand{\e}{{\rm e}} \ell\to\infty$ the spectrum contains only a finite number of complex-conjugate eigenvalues, which correspond to complex zeros k whose behavior is not covered by expansion (3.10).

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The third group of spectral singularities (green curves in figure 6(a)) was obtained by means of the continuation from the solution $\newcommand{\g}{\gamma} \gamma_\star^{(2)}$ and $k_\star^{(2)}$ in table 1. In view of the very complicated structure of the overall resulting picture, we only show a section of these curves corresponding to relatively small values of the gain-and-loss amplitudes $\newcommand{\g}{\gamma} \gamma$. Quite interestingly, there is no gap between the blue and green groups of curves, which results in the multitude of intersections between blue and green curves, see figure 6(a) and the magnified view in figure 6(c). These intersections suggest a possibility of simultaneous emergence of two complex-conjugate pairs of eigenvalues from two different interior points of the continuous spectra.

Considering further solutions $\newcommand{\g}{\gamma} \gamma_\star^{(n)}$, $k_\star^{(n)}$, $n=3,4,\ldots$, in the limit $\newcommand{\e}{{\rm e}} \ell=0$, one can construct new groups of spectral singularities with larger values of $\newcommand{\g}{\gamma} \gamma$, which are not shown in figure 6.