2012/03/31 by Mark S. Ashbaugh, Fritz Gesztesy, Marius Mitrea +2 · 1 citation
Mathematics · #math.SP #msc:35J25 #msc:35J40 #msc:35P15 #msc:35P05 #msc:46E35 #msc:47A10 #msc:47F05
paper · pdf · doi:10.1007/978-3-0348-0591-9_1
published as in Mathematical Physics, Spectral Theory and Stochastic Analysis, M. Demuth and W. Kirsch (eds.), 1-106, Oper. Theory Adv. Appl. 232, Birkhauser, Basel, 2013 · 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.1442
arxiv created 2014/01/20 · arxiv updated 2014/05/09
In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, S≥ ε IH for some ε >0 in a Hilbert space H to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for HK,Ω, the Krein--von Neumann extension of the perturbed Laplacian -Δ+V (in short, the perturbed Krein Laplacian) defined on C^∞0(Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂ℝn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r>1/2.