Spectra of Schrödinger operators on planar domains with ends with unbounded cross-section
✍ Scribed by Dr. Julian Edward
- Publisher
- John Wiley and Sons
- Year
- 1999
- Tongue
- English
- Weight
- 210 KB
- Volume
- 22
- Category
- Article
- ISSN
- 0170-4214
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✦ Synopsis
This paper studies the spectral theory of Schro¨dinger operators on planar domains with ends of increasing cross-section. We consider Dirichlet, Neumann, and certain mixed conditions, and the potentials » satisfy »"o(1) and P»"o( ), 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"(xN,, with p3(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"(xN,, with p3(0, 3#(8), Mourre Theory is applied to prove (i) the eigenvalues are of finite multiplicity and can accumulate only at 0 or R, (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.