A Survey on the Krein–von Neumann Extension, the Corresponding Abstract Buckling Problem, and Weyl-type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

Abstract

Abstract. 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, ≥ ℋ for some > 0 in a Hilbert space ℋ to an abstract buckling problem operator.In the concrete case where

Capacities, Removable Sets and Lp-Uniqueness on Wiener Spaces

“…Théorème] and [37], see also [2, Theorem 5.1.13]. Combined with Theorem 5.2 this yields a necessary codimension condition which is similar as in the case of Laplacians on Euclidean spaces, [5,25]. If (−(−L) m , F C ∞ b (N)) is L p -unique, then ̺ d (Σ) = 0 for all d < 2mp.…”

“…It is well known that this operator is essential self-adjoint in L 2 (R n ) if and only if n ≥ 4, [47, p.114] and [39,Theorem X.11,p.161]. Generalizations of this example to manifolds have been provided in [12] and [34], more general examples on Euclidean spaces can be found in [5] and [25], further generalizations to manifolds and metric measure spaces will be discussed in [26]. For the Laplacian on R n one main observation is that, if a compact set Σ of zero measure is removed from R n , the essential self-adjointness of (∆, C ∞ c (R n \ Σ)) in L 2 (R n ) implies that dim H Σ ≤ n − 4, where dim H denotes the Hausdorff dimension.…”

“…For the Laplacian on R n one main observation is that, if a compact set Σ of zero measure is removed from R n , the essential self-adjointness of (∆, C ∞ c (R n \ Σ)) in L 2 (R n ) implies that dim H Σ ≤ n − 4, where dim H denotes the Hausdorff dimension. See [5,Theorems 10.3 and 10.5] or [25,Theorem 2]. This necessary 'codimension four' condition can be rephrased by saying that we must have H n−d (Σ) = 0 for all d < 4, where H n−d denotes the Hausdorff measure of dimension n − d.…”

QFT Over the Finite Line. Heat Kernel Coefficients, Spectral Zeta Functions and Selfadjoint Extensions

“…Following subsection 11.1 in Ref. [23], the VNK extension of the operator T = − over the finite interval [0, L] is characterized as the unique selfadjoint extension with a maximal number of zero modes. From Equation (2.5), the maximum number of zero modes for the Laplace operator over the finite line is two: a constant zero mode and a linear zero mode.…”

Krein Extension of an Even-Order Differential Operator

“…In [5] the authors study spectral properties for H K,Ω , the Krein extension of the perturbed Laplacian −∆ + V defined on C ∞ 0 (Ω), where V is measurable, bounded and nonnegative, in a bounded open set Ω ⊂ R n belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C 1,r , r > 1/2. See also [3,4,7,11,21,24,25,26,33] and the references therein. However, the problem of finding M(0) is nontrivial even in the case of positively definite operator.…”