Bound states in waveguides with complex Robin boundary conditions

Abstract

We consider the Laplacian in a tubular neighbourhood of a hyperplane subjected to nonself-adjoint PT -symmetric Robin boundary conditions. Its spectrum is found to be purely essential and real for constant boundary conditions. The influence of the perturbation in the boundary conditions on the threshold of the essential spectrum is studied using the Birman-Schwinger principle. Our aim is to derive a sufficient condition for existence, uniqueness and reality of discrete eigenvalues. We show that discrete spectrum exists when the perturbation acts in the mean against the unperturbed boundary conditions and we are able to obtain the first term in its asymptotic expansion in the weak coupling regime.

Spectra of definite type in waveguide models

“…It appears that only (i) and moreover only in the simplest setting has been known, cf. [7,21]. As our approach does not rely on an explicit knowledge of Green's function for the unperturbed operator, we obtain analogous conclusions without additional efforts also if rather general regular potentials are included (in which case the picture of definite type spectra may be much richer, see Figure 3.3).…”

“…In the simplest setting of PT -symmetric waveguide (i.e. without potentials), see [7,6,15,21], we prove that the lowest part of the essential spectrum of the unperturbed waveguide is of positive type; cf. Theorem 3.6 and Figure 3.2.…”

“…We investigate the two-dimensional PT -symmetric waveguide suggested in [7] and studied further in [6,15,21]. We view the waveguide as a perturbation of a certain "unperturbed" operator defined in Subsection 3.2, having the tensor product structure (1.3).…”

“…We indicate how this slight extension of [7, Prop. 5.1] and [21,Prop. 4.7], where less general perturbations in boundary conditions were considered, can be proved.…”

On the discrete spectrum of Schrödinger operators with Ahlfors regular potentials in a strip

“…where α, β ∈ R. Here u x i (i = 1, 2) denotes the partial derivative of u with respect to x i and note that µ does not have to be the two-dimensional Lebesgue measure. A physical motivation to this problem is closely related to the study of spectral properties of quantum waveguides (see, e.g., [8,9,18,21,24]).…”