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Some quantum operators with discrete spectrum but classically continuous spectrum

✍ Scribed by Barry Simon

Publisher
Elsevier Science
Year
1983
Tongue
English
Weight
596 KB
Volume
146
Category
Article
ISSN
0003-4916

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✦ Synopsis

We consider a number of simple quantum Hamiltonians H(-iV, x) with the following property: H(-rV, x) has discrete spectrum even though {(p, q) 1 H(p, q) < E} has infinite volume.

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## Abstract We show that when a potential __b~n~__ of a discrete Schrödinger operator, defined on __l__^2^(ℤ^+^), slowly oscillates satisfying the conditions __b~n~__ ∈ __l__^∞^ and ∂__b~n~__ = __b__~__n__ +1~ – __b~n~__ ∈ __l^p^__, __p__ < 2, then all solutions of the equation __Ju__ = __Eu__ are