Milos Tater

Bound states and scattering in quantum waveguides coupled laterally through a boundary window

We consider a pair of parallel straight quantum waveguides coupled laterally through a window of a width l in the common boundary. We show that such a system has at least one bound state for any l≳0. We find the corresponding eigenvalues and eigenfunctions numerically using the mode‐matching method, and discuss their behavior in several situations. We also discuss the scattering problem in this setup, in particular, the turbulent behavior of the probability flow associated with resonances. The level and phase‐shift spacing statistics shows that in distinction to closed pseudo‐integrable billiards, the present system is essentially nonchaotic. Finally, we illustrate time evolution of wave packets in the present model.

On spectral polynomials of the Heun equation. I

The classical Heun equation has the form {Q(z)d^2dz^2+P(z)ddz+V(z)}S(z)=0, where Q(z) is a cubic complex polynomial, P(z) is a polynomial of degree at most 2 and V(z) is at most linear. In the second half of the nineteenth century E. Heine and T. Stieltjes initiated the study of the set of all V(z) for which the above equation has a polynomial solution S(z) of a given degree n. The main goal of the present paper is to study the union of the roots of the latter set of V(z)s when n->~. We provide an explicit description of this limiting set and give a substantial amount of preliminary and additional information about it obtained using a certain technique developed by A.B.J. Kuijlaars and W. Van Assche.

Spectrum of Dirichlet Laplacian in a conical layer

We study the spectral properties of Dirichlet Laplacian on the conical layer of the opening angle π − 2θ and thickness equal to π. We demonstrate that below the continuum threshold, which is equal to 1, there is an infinite sequence of isolated eigenvalues and analyse properties of these geometrically induced bound states. By numerical computation we find examples of the eigenfunctions.

A single-mode quantum transport in serial-structure geometric scatterers

We study transport in quantum systems consisting of a finite array of N identical single-channel scatterers. A general expression of the S matrix in terms of the individual-element data obtained recently for potential scattering is rederived in this wider context. It shows in particular how the band spectrum of the infinite periodic system arises in the limit N→∞. We illustrate the result on two kinds of examples. The first are serial graphs obtained by chaining loops or T-junctions. Another example concerns geometric scatterers where the individual element consists of a surface with a pair of leads; we show that apart from the resonances coming from the decoupled-surface eigenvalues, such scatterers exhibit the high-energy behavior typical for the δ′ interaction for the physically interesting couplings.

Spectra of soft ring graphs

We discuss a ring-shaped soft quantum wire modelled by δ interaction supported by the ring with a generally nonconstant coupling strength. We derive the condition which determines the discrete spectrum of such systems, and analyse the dependence of the eigenvalues and eigenfunctions on the coupling and ring geometry. In particular, we illustrate that a random component in the coupling leads to a localization. The discrete spectrum is also investigated in the situation when the ring is placed into a homogeneous magnetic field or threaded by an Aharonov–Bohm flux and the system exhibits persistent currents. (Some figures in this article are in colour only in the electronic version)

Non-Hermitian spectral effects in a PT-symmetric waveguide

We present a numerical study of the spectrum of the Laplacian in an unbounded strip with PT-symmetric boundary conditions. We focus on non-Hermitian features of the model reflected in an unusual dependence of the eigenvalues below the continuous spectrum on various boundary-coupling parameters.

Evanescent modes in a multiple scattering factorization

We discuss differences between the exactS-matrix for scattering on serial structures and a known factorized expression constructed of single-elementS-matrices. As an illustration, we use an exactly solvable model of a quantum wire with two point impurities.

Exactly solvable three-body Calogero-type model with translucent two-body barriers

Abstract A new exactly solvable alternative to the Calogero three-particle model is proposed. Sharing its confining long-range part, it contains the mere zero-range two-particle barriers. Their penetrability gives rise to a tunneling, tunable via their three independent strengths. Their variability can control the removal of the degeneracy of the energy levels in an innovative, non-perturbative manner.

On resonances and bound states of Smilansky Hamiltonian

We consider the self-adjoint Smilansky Hamiltonian

Spectral analysis of a class of Schrödinger operators exhibiting a parameter-dependent spectral transition

\mathsf{H}_\varepsilon