Abstract
Consider the Schrödinger operator −Δ+V in L 2(ℝn) with a potential V, locally integrable and semibounded below. As we mentioned in Sect. 16.6, Molchanov’s criterion (16.6.2) involves the so-called negligible sets F, that is, sets of sufficiently small harmonic capacity.
In Sects. 18.2–18.3 we show that the constant c n given by (16.6.4) can be replaced by an arbitrary constant γ, 0<γ<1. We even establish a stronger result allowing negligibility conditions with γ depending on d and completely describe all admissible functions γ. More precisely, in the necessary condition for the discreteness of spectrum we allow arbitrary functions γ:(0,+∞)→(0,1). If γ(d)=O(d 2) in the negligibility condition (16.6.3), then it fails to be sufficient, i.e., it may happen that it is satisfied but the spectrum is not discrete (Sect. 18.4). However, we show that in the sufficient condition we can admit arbitrary functions γ with values in (0,1), defined for d>0 in a neighborhood of d=0 and satisfying
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© 2011 Springer-Verlag Berlin Heidelberg
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Maz’ya, V. (2011). Spectrum of the Schrödinger Operator and the Dirichlet Laplacian. In: Sobolev Spaces. Grundlehren der mathematischen Wissenschaften, vol 342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15564-2_18
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DOI: https://doi.org/10.1007/978-3-642-15564-2_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15563-5
Online ISBN: 978-3-642-15564-2
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