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Estimates for Periodic and Dirichlet Eigenvalues of the Schrödinger Operator with Singular Potentials

✍ Scribed by T. Kappeler; C. Möhr

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
210 KB
Volume
186
Category
Article
ISSN
0022-1236

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✦ Synopsis

In this paper, the periodic and the Dirichlet problems for the Schrödinger operator -(d 2 /dx 2 )+V are studied for singular, complex-valued potentials V in the Sobolev space H -a per [0, 1] (0 [ a < 1). The following results are shown:

(1) The periodic spectrum consists of a sequence (l k ) k \ 0 of complex eigenvalues satisfying the asymptotics (for any e > 0)

where V ˆ(k) denote the Fourier coefficients of V.

(2) The Dirichlet spectrum consists of a sequence (m n ) n \ 1 of complex eigenvalues satisfying the asymptotics (for any e > 0) m n =n 2 p 2 +V ˆ(0) -V ˆ(-2n)+V ˆ(2n) 2 +O(n 2a -1+e ).

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