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Minimal potential results for the Schrödinger equation in a slab

  • Laura De Carli , Steve Hudson and Xiaosheng Li EMAIL logo
Published/Copyright: July 15, 2015

Abstract

Consider the Schrödinger equation -Δu=(k+V)u in an infinite slab S=n-1×(0,1), where V has some type of decay at infinity; three specific types are studied. For each type, we prove necessary conditions for the existence of nontrivial admissible solutions. These conditions involve the norm of V, and the distance of k from the set 𝒦={π2m2:m}. If VL(S) is supported on a set D of finite measure, then these conditions also involve the measure of D, and, in many cases, the inequalities are sharp.

Award Identifier / Grant number: DMS-1109561

Funding statement: Xiaosheng Li is supported in part by NSF grant DMS-1109561.

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Received: 2014-4-28
Revised: 2014-9-12
Published Online: 2015-7-15
Published in Print: 2016-7-1

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