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RobintoRobin Maps and KreinType Resolvent Formulas for Schrödinger Operators on Bounded Lipschitz Domains
, 2008
"... We study RobintoRobin maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2, with generalized Robin boundary conditions. ..."
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Cited by 28 (10 self)
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We study RobintoRobin maps, and Kreintype resolvent formulas for Schrödinger operators on bounded Lipschitz domains in R n, n � 2, with generalized Robin boundary conditions.
Kreins resolvent formula for selfadjoint extensions of symmetric secondorder elliptic differential operators
 J. Phys. A: Math. Theor
, 2009
"... Abstract. Given a symmetric, semibounded, second order elliptic differential operator A on a bounded domain with C 1,1 boundary, we provide a Kreĭntype formula for the resolvent difference between its Friedrichs extension and an arbitrary selfadjoint one. 1. Introduction. Given a bounded open set ..."
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Cited by 24 (2 self)
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Abstract. Given a symmetric, semibounded, second order elliptic differential operator A on a bounded domain with C 1,1 boundary, we provide a Kreĭntype formula for the resolvent difference between its Friedrichs extension and an arbitrary selfadjoint one. 1. Introduction. Given a bounded open set Ω ⊂ Rn, n> 1, let us consider a second order elliptic differential operator A: C ∞ c (Ω) ⊂ L 2 (Ω) → L 2 n∑ n∑
Generalized Qfunctions and DirichlettoNeumann maps for elliptic differential operators
"... Abstract. The classical concept of Qfunctions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the DirichlettoNeumann map in the theory of elliptic differential equations can be interpreted as a generalized Qfunction. For couplings ..."
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Cited by 19 (8 self)
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Abstract. The classical concept of Qfunctions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the DirichlettoNeumann map in the theory of elliptic differential equations can be interpreted as a generalized Qfunction. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an H 2framework are obtained. 1.
Spectral theory for perturbed Krein Laplacians in nonsmooth domains
, 2010
"... We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V 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 do ..."
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Cited by 16 (11 self)
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We study spectral properties for HK,Ω, the Krein–von Neumann extension of the perturbed Laplacian − ∆ + V 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. In particular, in the aforementioned context we establish the Weyl asymptotic formula #{j ∈ N  λK,Ω,j ≤ λ} = (2π) −n vnΩ  λ n/2 + O ` λ (n−(1/2))/2 ´ as λ → ∞, where vn = πn/2 /Γ((n/2)+1) denotes the volume of the unit ball in Rn, and λK,Ω,j, j ∈ N, are the nonzero eigenvalues of HK,Ω, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein–von Neumann extension of − ∆ + V defined on C ∞ 0 (Ω)) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980’s. Our work builds on that of Grubb in the early 1980’s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting. We also study certain exteriortype domains Ω = Rn \K, n ≥ 3, with K ⊂ Rn compact and vanishing
NONLOCAL ROBIN LAPLACIANS AND SOME REMARKS ON A PAPER BY FILONOV
, 2009
"... The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to selfadjoint Laplacians −∆Θ,Ω in L 2 (Ω; d n x) with (nonlocal and local) Robintype boundary conditions on bounded Lipschitz domains Ω ⊂ R n, n ∈ N, n ≥ 2. Second, we e ..."
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Cited by 16 (5 self)
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The aim of this paper is twofold: First, we characterize an essentially optimal class of boundary operators Θ which give rise to selfadjoint Laplacians −∆Θ,Ω in L 2 (Ω; d n x) with (nonlocal and local) Robintype boundary conditions on bounded Lipschitz domains Ω ⊂ R n, n ∈ N, n ≥ 2. Second, we extend Friedlander’s inequalities between Neumann and Dirichlet Laplacian eigenvalues to those between nonlocal Robin and Dirichlet Laplacian eigenvalues associated with bounded Lipschitz domains Ω, following an approach introduced by Filonov for this type of problems.
A description of all selfadjoint extensions of the Laplacian and Krĕıntype resolvent formulas on nonsmooth domains
 J. Anal. Math
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Singular integrals and elliptic boundary problems on regular SemmesKenigToro domains
, 2008
"... We develop the theory of layer potentials and related singular integral operators as a tool to study a variety of elliptic boundary problems on a family of domains introduced by Semmes [101]–[102] and Kenig and Toro [64]–[66], which we call regular SemmesKenigToro domains. This extends the classic ..."
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Cited by 14 (4 self)
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We develop the theory of layer potentials and related singular integral operators as a tool to study a variety of elliptic boundary problems on a family of domains introduced by Semmes [101]–[102] and Kenig and Toro [64]–[66], which we call regular SemmesKenigToro domains. This extends the classic work of Fabes, Jodeit, and Rivière in several ways. For one, the class of domains considered contains the class of VMO1 domains, which in turn contains the class of C 1 domains. In addition we study not only the Dirichlet and Neumann boundary problems, but also a variety of others. Furthermore, we treat not only constant coefficient operators, but also operators with variable coefficients, including operators on manifolds. Contents 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 13 (9 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 abstract Titchmarsh–Weyl Mfunction for adjoint operator pairs and its relation to the spectrum
 INTEGRAL EQ. OPERATOR TH.
, 2009
"... In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl Mfunction see the same singularities as the resolvent of a certain restriction AB of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces S and ˜ ..."
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Cited by 11 (0 self)
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In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl Mfunction see the same singularities as the resolvent of a certain restriction AB of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces S and ˜ S such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the Mfunction is analytic. We present four examples – one involving a HainLüst type operator, one involving a perturbed Friedrichs operator and two involving a simple ordinary differential operators on a half line – which together indicate that the abstract results are probably best possible.