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On the Spectrum of the Dirac Operator and the Existence of Discrete Eigenvalues for the Defocusing Nonlinear Schrödinger Equation
Author(s) -
Biondini Gino,
Prinari Barbara
Publication year - 2014
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12024
Subject(s) - eigenvalues and eigenvectors , mathematics , spectrum (functional analysis) , mathematical analysis , operator (biology) , infinity , piecewise , nonlinear system , constant (computer programming) , dirac (video compression format) , mathematical physics , essential spectrum , dirac operator , boundary value problem , dirac equation , measure (data warehouse) , continuous spectrum , nonlinear schrödinger equation , schrödinger equation , quantum mechanics , physics , biochemistry , chemistry , repressor , database , computer science , transcription factor , neutrino , gene , programming language
We revisit the scattering problem for the defocusing nonlinear Schrödinger equation with constant, nonzero boundary conditions at infinity, i.e., the eigenvalue problem for the Dirac operator with nonzero rest mass. By considering a specific kind of piecewise constant potentials we address and clarify two issues, concerning: (i) the (non)existence of an area theorem relating the presence/absence of discrete eigenvalues to an appropriate measure of the initial condition; and (ii) the existence of a contribution to the asymptotic phase difference of the potential from the continuous spectrum.