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On the structure of the essential spectrum for the three‐particle Schrödinger operators on lattices
Author(s) -
Albeverio Sergio,
Lakaev Saidakhmat N.,
Muminov Zahriddin I.
Publication year - 2007
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200410509
Subject(s) - essential spectrum , mathematics , operator (biology) , bounded function , eigenvalues and eigenvectors , schrödinger's cat , discrete spectrum , spectrum (functional analysis) , hilbert space , lattice (music) , momentum (technical analysis) , bounded operator , momentum operator , energy operator , ladder operator , quantum mechanics , mathematical physics , pure mathematics , mathematical analysis , compact operator , physics , energy (signal processing) , repressor , chemistry , computer science , acoustics , biochemistry , transcription factor , programming language , finance , economics , extension (predicate logic) , gene , statistics
A system of three quantum particles on the three‐dimensional lattice ℤ 3 with arbitrary dispersion functions having not necessarily compact support and interacting via short‐range pair potentials is considered. The energy operators of the systems of the two‐and three‐particles on the lattice ℤ 3 in the coordinate and momentum representations are described as bounded self‐adjoint operators on the corresponding Hilbert spaces. For all sufficiently small values of the two‐particle quasi‐momentum k ∈ (– π , π ] 3 the finiteness of the number of eigenvalues of the two‐particle discrete Schrödinger operator h α ( k ) below the continuous spectrum is established. The location of the essential spectrum of the three‐particle discrete Schrödinger operator H ( K ), K ∈ (– π , π ] 3 being the three‐particle quasi‐momentum, is described by means of the spectrum of the two‐particle discrete Schrödinger operator h α ( k ), k ∈ (– π , π ] 3 . It is established that the essential spectrum of the three‐particle discrete Schrödinger operator H ( K ), K ∈ (– π , π ] 3 , consists of finitely many bounded closed intervals. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)