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WEYLTITCHMARSH THEORY FOR STURMLIOUVILLE OPERATORS WITH DISTRIBUTIONAL COEFFICIENTS
, 2012
"... We systematically develop Weyl–Titchmarsh theory for singular differential operators on arbitrary intervals (a,b) ⊆ R associated with rather general differential expressions of the type τf = 1 ..."
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Cited by 7 (6 self)
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We systematically develop Weyl–Titchmarsh theory for singular differential operators on arbitrary intervals (a,b) ⊆ R associated with rather general differential expressions of the type τf = 1
Selfadjoint extensions of the Laplacian and Kreintype resolvent formulas in nonsmooth domains
, 2009
"... This paper has two main goals. First, we are concerned with the classification of selfadjoint extensions of the Laplacian − ∆ ˛ ˛ C ∞ 0 (Ω) in L2 (Ω; d n x). Here, the domain Ω belongs to a subclass of bounded Lipschitz domains (which we term quasiconvex domains), which contain all convex domai ..."
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Cited by 6 (5 self)
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This paper has two main goals. First, we are concerned with the classification of selfadjoint extensions of the Laplacian − ∆ ˛ ˛ C ∞ 0 (Ω) in L2 (Ω; d n x). Here, the domain Ω belongs to a subclass of bounded Lipschitz domains (which we term quasiconvex domains), which contain all convex domains, as well as all domains of class C 1,r, for r ∈ (1/2, 1). Second, we establish Kreintype formulas for the resolvents of the various selfadjoint extensions of the Laplacian in quasiconvex domains and study the properties of the corresponding Weyl–Titchmarsh operators (or energydependent DirichlettoNeumann maps). One significant technical innovation in this paper is an extension of the classical boundary trace theory for functions in spaces which lack Sobolev regularity in a traditional sense, but are suitably adapted to the
The Krein–von Neumann extension and its connection to an abstract buckling problem
 Math. Nachr
, 2010
"... Abstract. 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 ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete cas ..."
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Cited by 6 (5 self)
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Abstract. 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 ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S = −∆  C ∞ 0 (Ω) in L2 (Ω; dnx) for Ω ⊂ Rn an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ̸ = 0, is in onetoone correspondence with the problem of the buckling of a clamped plate, (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0 (Ω), where u and v are related via the pair of formulas u = S −1 F (−∆)v, v = λ−1 (−∆)u, with SF the Friedrichs extension of S. 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.). 1.
CONNECTION TO AN ABSTRACT BUCKLING PROBLEM
"... Dedicated to the memory of Erhard Schmidt (1876–1959). Abstract. 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 ≥ εIH for some ε> 0 in a Hilbert space H to an ..."
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Dedicated to the memory of Erhard Schmidt (1876–1959). Abstract. 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 ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S = −∆  C ∞ 0 (Ω) in L 2 (Ω; d n x) for Ω ⊂ R n an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ̸ = 0, is in onetoone correspondence with the problem of the buckling of a clamped plate, (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0(Ω), where u and v are related via the pair of formulas u = S −1 F (−∆)v, v = λ−1 (−∆)u, with SF the Friedrichs extension of S. 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.). 1.
CORRESPONDING ABSTRACT BUCKLING PROBLEM, AND WEYLTYPE SPECTRAL ASYMPTOTICS FOR PERTURBED KREIN LAPLACIANS IN NONSMOOTH DOMAINS
, 2012
"... Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. 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 ..."
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Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. 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, S ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S = −∆  C ∞ 0 (Ω) in L 2 (Ω;d n x) for Ω ⊂ R n an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ̸ = 0, is in onetoone correspondence with the problem of the buckling of a clamped plate, where u and v are related via the pair of formulas (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0(Ω), u = S −1 F (−∆)v, v = λ−1 (−∆)u,
A SURVEY ON THE KREIN–VON NEUMANN EXTENSION, THE CORRESPONDING ABSTRACT BUCKLING PROBLEM, AND WEYLTYPE SPECTRAL ASYMPTOTICS FOR PERTURBED KREIN LAPLACIANS IN NONSMOOTH DOMAINS
"... Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. 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 ..."
Abstract
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Dedicated with great pleasure to Michael Demuth on the occasion of his 65th birthday. 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, S ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S = −∆  C ∞ 0 (Ω) in L2 (Ω; dnx) for Ω ⊂ Rn an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ̸ = 0, is in onetoone correspondence with the problem of the buckling of a clamped plate, where u and v are related via the pair of formulas (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0 (Ω), u = S −1 F (−∆)v, v = λ−1 (−∆)u, with SF the Friedrichs extension of S.
Spectral Theory as Influenced by Fritz Gesztesy
, 2012
"... Abstract. We survey a selection of Fritz’s principal contributions to the field of spectral theory and, in particular, to Schrödinger operators. 1. ..."
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Abstract. We survey a selection of Fritz’s principal contributions to the field of spectral theory and, in particular, to Schrödinger operators. 1.
The Kreinvon Neumann . . . CONNECTION TO AN ABSTRACT BUCKLING PROBLEM
, 2009
"... 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 ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S ..."
Abstract
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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 ≥ εIH for some ε> 0 in a Hilbert space H to an abstract buckling problem operator. In the concrete case where S = −∆  C ∞ 0 (Ω) in L 2 (Ω; d n x) for Ω ⊂ R n an open, bounded (and sufficiently regular) domain, this recovers, as a particular case of a general result due to G. Grubb, that the eigenvalue problem for the Krein Laplacian SK (i.e., the Krein–von Neumann extension of S), SKv = λv, λ ̸ = 0, is in onetoone correspondence with the problem of the buckling of a clamped plate, (−∆) 2 u = λ(−∆)u in Ω, λ ̸ = 0, u ∈ H 2 0(Ω), where u and v are related via the pair of formulas u = S −1 F (−∆)v, v = λ−1 (−∆)u, with SF the Friedrichs extension of S. 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.).