Spectra of Schrödinger Operators on Planar Domains with Ends with Unbounded Cross-section Article

cited authors

  • Edward, J

fiu authors

abstract

  • This paper studies the spectral theory of Schrodinger 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| < xp}, 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| < xp}, 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.

publication date

  • January 25, 1999

start page

  • 139

end page

  • 169

volume

  • 22

issue

  • 2