The spectrum of discrete Dirac operator with a general boundary condition

Nimet Coskun,Nihal Yokus

doi:10.1186/s13662-020-02851-2

Nimet Coskun, Nihal Yokus

Open Access

https://doi.org/10.1186/s13662-020-02851-2

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Abstract

In this paper, we aim to investigate the spectrum of the nonselfadjoint operator L generated in the Hilbert space l_{2}(mathbb{N},mathbb{C}^{2}) by the discrete Dirac system{yn+1(2)−yn(2)+pnyn(1)=λyn(1),−yn(1)+yn−1(1)+qnyn(2)=λyn(2),n∈N,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} y_{n+1}^{ (2 )} - y_{n}^{ (2 )} + p_{n} y_{n}^{ (1 )} =\\lambda y_{n}^{ (1 )},\\\\ - y_{n}^{ (1 )} + y_{n-1}^{ (1 )} + q_{n} y_{n}^{ (2 )} =\\lambda y_{n}^{ (2 )}, \\end{cases}\\displaystyle \\quad n\\in \\mathbb{N}, $$\\end{document} and the general boundary condition∑n=0∞hnyn=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\sum_{n = 0}^{\\infty } h_{n}y_{n} = 0, $$\\end{document} where λ is a spectral parameter, Δ is the forward difference operator, (h_{n}) is a complex vector sequence such that h_{n} = ( h_{n}^{(1)}, h_{n}^{(2)} ), where h_{n}^{(i)} in l^{1} ( mathbb{N} ) cap l^{2} ( mathbb{N} ), i = 1,2, and h_{0}^{(1)} ne 0. Upon determining the sets of eigenvalues and spectral singularities of L, we prove that, under certain conditions, L has a finite number of eigenvalues and spectral singularities with finite multiplicity.

Highlights

Along with the invention of the Schrödinger equation, the physical scope of mathematical problems connected with the spectra of differential equations with prescribed boundary conditions was enormously enlarged
We aim to investigate the spectrum of the nonselfadjoint operator L generated in the Hilbert space l2(N, C2) by the discrete Dirac system yn(2+)1 – yn(2) + pnyn(1) = λyn(1), –yn(1) + yn(1–)1 + qnyn(2) = λyn(2), n ∈ N, and the general boundary condition hnyn = 0, n=0 where λ is a spectral parameter, is the forward difference operator, is a complex vector sequence such that hn = (h(n1), h(n2)), where h(ni) ∈ l1(N) ∩ l2(N), i = 1, 2, and h(01) = 0
Upon determining the sets of eigenvalues and spectral singularities of L, we prove that, under certain conditions, L has a finite number of eigenvalues and spectral singularities with finite multiplicity

Summary

Introduction

Along with the invention of the Schrödinger equation, the physical scope of mathematical problems connected with the spectra of differential equations with prescribed boundary conditions was enormously enlarged. The types of equations that previously had applications only to mechanical vibrations were to be used for the description of atoms and molecules. There are important and altogether astonishing applications of the results obtained in the spectral theory of linear operators in Hilbert spaces to scattering theory, inverse problems, and quantum mechanics. The Hamiltonian of a quantum particle confined to a box involves a choice of boundary conditions at the box ends. Since different choices of boundary conditions imply different physical models, spectral theory of operators with boundary conditions constitues a progressing field of investigation [1, 2]

Let T denote a matrix operator

Let us define the sets

For β

Im z