Premium
Spectra of Schrödinger operators on planar domains with ends with unbounded cross‐section
Author(s) -
Edward Julian
Publication year - 1999
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19990125)22:2<139::aid-mma29>3.0.co;2-h
Subject(s) - mathematics , eigenvalues and eigenvectors , planar , multiplicity (mathematics) , spectrum (functional analysis) , limiting , section (typography) , dirichlet distribution , pure mathematics , mathematical analysis , spectral theory , mathematical physics , quantum mechanics , physics , boundary value problem , hilbert space , mechanical engineering , computer graphics (images) , computer science , advertising , engineering , business
This paper studies the spectral theory of Schrödinger operators on planar domains with ends of increasing cross‐section. We consider Dirichlet, Neumann, and certain mixed conditions, and the potentials V satisfy V = o (1) and PV = o (1), where P is a certain vector field determined by the geometry of the end. For a class of domains which includes {( x, y ); x > 1, ∣ y ∣ < x p }, with p ∈ (1/2, 2), the absence of positive eigenvalues is proved. The proof is an adaptation of a method of Kato. For another class of domains which includes {( x, y ); x > 1, ∣ y ∣ < x p }, with p ∈ (0, 3 + √8), Mourre Theory is applied to prove (i) the eigenvalues are of finite multiplicity and can accumulate only at 0 or ∞, (ii) there is no singular continuous spectrum, and (iii) a Limiting Absorption Principle holds away from 0 and the eigenvalues. Under weaker hypotheses on the potential, the results above are shown to hold at higher energies. Copyright © 1999 John Wiley & Sons, Ltd.