Abstract
We establish necessary and sufficient conditions for localized complex potentials in the Schrödinger equation to enable spectral singularities (SSs) and show that such potentials have the universal form , where w(x) is a differentiable function, such that , and k0 is a non-zero real. We also find that when k0 is a complex number, then the eigenvalue of the corresponding Shrödinger operator has an exact solution which, depending on k0, represents a coherent perfect absorber (CPA), laser, a localized bound state, a quasi bound state in the continuum (a quasi-BIC), or an exceptional point (the latter requiring additional conditions). Thus, k0 is a bifurcation parameter that describes transformations among all those solutions. Additionally, in a more specific case of a real-valued function w(x) the resulting potential, although not being symmetric, can feature a self-dual SS associated with the CPA-laser operation. At this moment, the complex potential has exactly two coexisting SSs at different points of the continuous spectrum. In the space of the system parameters, the transition through each self-dual SS corresponds to a bifurcation of a pair of complex-conjugate propagation constants from the continuum. The bifurcation of a first complex-conjugate pair corresponds to the phase transition from purely real to complex spectrum.
Highlights
Singularities of the spectral characteristics of non-Hermitian operators, alias spectral singularities (SSs), were introduced in mathematical literature more than six decades ago [1] and were well studied since [2, 3, 4]
In a more specific situation of real-valued function w(x) we demonstrate that the found SS solution can coexist with another, self-dual spectral singularity which corresponds to the combined coherent perfect absorber (CPA)-laser operation
Ψ(x) = 0 for x ∈ R, neither w(x) nor U(x) have singularities. In this last case, solving (8) with respect to ψ0, we find that the solution corresponding to the spectral singularity of potential (12) can be expressed directly through the function w(x), as x ψ0(x) = ρ exp −i w(ξ)dξ, (13)
Summary
Singularities of the spectral characteristics of non-Hermitian operators, alias spectral singularities (SSs), were introduced in mathematical literature more than six decades ago [1] and were well studied since [2, 3, 4]. While the definition of a SS can be formulated in terms of poles of a truncated resolvent of a non-Hermitian Schrodinger operator in any spatial dimension (see e.g. the discussion in [15] and references therein), in this work we deal only with one-dimensional setting In this case a convenient description of the SSs can be elaborated in terms of the transfer matrix M(k) depending on the wavenumber k, when real zeros of the matrix element M22(k) determine SSs [9, 10].
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