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Discreteness of spectrum and positivity criteria for Schroedinger operators
Ohio State Univ, Columbus, OH 43210 USA.
Northeastern Univ, Boston, MA 02115 USA .
2005 (English)In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 162, no 2Article in journal (Refereed) Published
Abstract [en]

We provide a class of necessary and sufficient conditions for the discreteness of spectrum of Schrodinger operators with scalar potentials which are semibounded below. The classical discreteness of spectrum criterion by A. M. Molchanov (1953) uses a notion of negligible set in a cube as a set whose Wiener capacity is less than a small constant times the capacity of the cube. We prove that this constant can be taken arbitrarily between 0 and 1. This solves a problem formulated by I. M. Gelfand in 1953. Moreover, we extend the notion of negligibility by allowing the constant to depend on the size of the cube. We give a complete description of all negligibility conditions of this kind. The a priori equivalence of our conditions involving different negligibility classes is a nontrivial property of the capacity. We also establish similar strict positivity criteria for the Schrodinger operators with nonnegative potentials.

Place, publisher, year, edition, pages
2005. Vol. 162, no 2
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-30259Local ID: 15769OAI: oai:DiVA.org:liu-30259DiVA: diva2:251081
Available from: 2009-10-09 Created: 2009-10-09 Last updated: 2017-12-13

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Maz´ya, Vladimir G.

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