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Nondassical eigenvalue distribution of one-dimensional Schrödinger operators

Published online by Cambridge University Press:  14 November 2011

M. Klaus
M. Klaus
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A.
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Synopsis

We consider differential operators of the form H = −d2/dx2 + q(x) acting on u ∈ L2(0,∞) with boundary condition u(0) = 0. The potential q(x) is such that H has essential spectrum [0,∞) and an infinite sequence of negative eigenvalues converging to zero. Let n(E) denote the number of eigenvalues of H which are less than E. Under certain conditions on q(x), the well-known formula n(E)∼(2φ)−1 vol {x, p | p2 + q(x)<E}, E↑0, holds. We shall study the validity of this formula for potentials which show oscillatory behaviour as x →∞, like e.g. q(x) = −(1 + x)−α(a + b sin x) with 0<α <2, a≧0, b≠0. We shall obtain the leading-order behaviour of both n(E) and vol n(E)∼(2φ)−1 vol {x, p | p2 + q(x)<E} as E↑0 for a certain class of q's, and we shall see that the classical formula fails in most cases, but there are some noteworthy exceptions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 101 , Issue 1-2 , 1985 , pp. 149 - 158
Copyright
Copyright © Royal Society of Edinburgh 1985

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