Numerical calculation of the discrete spectra of one‐dimensional Schrödinger operators with point interactions

Abstract

In this paper, we consider one‐dimensional Schrödinger operators Sq on double-struckR with a bounded potential q supported on the segment []h0,h1 and a singular potential supported at the ends h0, h1. We consider an extension of the operator Sq in L2()R defined by the Schrödinger operator Hq=−d2dx2+q and matrix point conditions at the ends h0, h1. By using the spectral parameter power series method, we derive the characteristic equation for calculating the discrete spectra of operator Hq. Moreover, we provide closed‐form expressions for the eigenfunctions and associate functions in the Jordan chain given in the form of power series of the spectral parameter. The validity of our approach is proven in several numerical examples including self‐adjoint and nonself‐adjoint problems involving general point interactions described in terms of δ‐ and δ′‐distributions.

On the calculation of the discrete spectra of one‐dimensional Dirac operators

“…Unlike the cited papers, the present work is devoted to the construction of the dispersion equations for the one‐dimensional Dirac operators, the zeros of which are eigenvalues of the discrete spectrum of the Dirac operators with singular and regular potentials. For the construction of the dispersion equations, we use the spectral parameter power series (SPPS for short) method which has been used earlier 33–35 for the study of the discrete spectra of one‐dimensional Schrödinger operators with singular potentials. In these cited works, the Schrödinger operator $$$$ {\mathcal{S}}_{q+{q}_s}=-\frac{d^2}{d{x}^2}+q(x)+{q}_s(x) $$$$ was equipped by potentials including a $$$ {L}^{\infty}\left(\mathbb{R}\right) $$$‐regular part $$$ q(x) $$$ and a singular part $$$ {q}_s(x) $$$ involving $$$ \delta $$$‐ and $$$ {\delta}^{\prime } $$$ interactions $$$$ {q}_s(x)=\sum \limits_{j=1}^N{\alpha}_j\delta \left(x-{x}_j\right)+{\beta}_j{\delta}^{\prime}\left(x-{x}_j\right).$$…”

“…Unlike the cited papers, the present work is devoted to the construction of the dispersion equations for the one-dimensional Dirac operators, the zeros of which are eigenvalues of the discrete spectrum of the Dirac operators with singular and regular potentials. For the construction of the dispersion equations, we use the spectral parameter power series (SPPS for short) method which has been used earlier [33][34][35] for the study of the discrete spectra of one-dimensional Schrödinger operators with singular potentials. In these cited works, the Schrödinger operator…”