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73
Stollmann: Eigenfunction expansion for Schrödinger operators on metric graphs (Preprint arXiv:0801.1376
"... Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices. ..."
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Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.
Asymptotic first eigenvalue estimates for the biharmonic operator on a rectangle, preprint
, 1996
"... We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper b ..."
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We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions. These bounds can be used to show that the negative part of the groundstate is small. 1.
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 7 (4 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Surface States and Spectra
, 2000
"... Let Z d+1 + = Z d Z+,letH 0 be the discrete Laplacian on the Hilbert space l 2 (Z d+1 + ) with a Dirichlet boundary condition, and let V be a potential supported on the boundary #Z d+1 + .We introduce the notions of surface states and surface spectrum of the operator H = H 0 + V and ex ..."
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Cited by 6 (1 self)
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Let Z d+1 + = Z d Z+,letH 0 be the discrete Laplacian on the Hilbert space l 2 (Z d+1 + ) with a Dirichlet boundary condition, and let V be a potential supported on the boundary #Z d+1 + .We introduce the notions of surface states and surface spectrum of the operator H = H 0 + V and explore their properties. Our main result is that if the potential V is random and if the disorder is either large or small enough, then in dimension two H has no surface spectrum on #(H 0 ) with probability one. To prove this result we combine AizenmanMolchanov theory with techniques of scattering theory.
Generating spectral gaps by geometry
, 2004
"... Abstract. Motivated by the analysis of Schrödinger operators with periodic potentials we consider the following abstract situation: Let ∆X be the Laplacian on a noncompact Riemannian covering manifold X with a discrete isometric group Γ acting on it such that the quotient X/Γ is a compact manifold. ..."
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Cited by 5 (4 self)
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Abstract. Motivated by the analysis of Schrödinger operators with periodic potentials we consider the following abstract situation: Let ∆X be the Laplacian on a noncompact Riemannian covering manifold X with a discrete isometric group Γ acting on it such that the quotient X/Γ is a compact manifold. We prove the existence of a finite number of spectral gaps for the operator ∆X associated with a suitable class of manifolds X with nonabelian covering transformation groups Γ. This result is based on the nonabelian Floquet theory as well as the MinMaxprinciple. Groups of type I specify a class of examples satisfying the assumptions of the main theorem. 1.
COMMUTATORS, EIGENVALUE GAPS, AND MEAN CURVATURE IN THE THEORY OF SCHRÖDINGER OPERATORS
, 2005
"... Commutator relations are used to investigate the spectra of Schrödinger Hamiltonians, H = − ∆ + V (x) , acting on functions of a smooth, compact ddimensional manifold M immersed in R d+1. Here ∆ denotes the LaplaceBeltrami operator, and the realvalued potential–energy function V (x) acts by multi ..."
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Commutator relations are used to investigate the spectra of Schrödinger Hamiltonians, H = − ∆ + V (x) , acting on functions of a smooth, compact ddimensional manifold M immersed in R d+1. Here ∆ denotes the LaplaceBeltrami operator, and the realvalued potential–energy function V (x) acts by multiplication. The manifold M may be complete or it may have a boundary, in which case Dirichlet boundary conditions are imposed. It is found that the mean curvature of a manifold poses tight constraints on the spectrum of H. Further, a special algebraic rôle is found to be played by a Schrödinger operator with potential proportional to the square of the mean curvature: Hg: = − ∆ + gh 2, where g is a real parameter and h:= d� κj, j=1 with {κj}, j = 1,..., d denoting the principal curvatures of M. For instance, by Theorem 2.1 and Corollary 3.4, each eigenvalue gap of an arbitrary Schrödinger operator is bounded above by an expression using H 1/4. The “isoperimetric ” parts of these theorems state that these bounds are sharp for the fundamental eigenvalue gap and for infinitely many other eigenvalue gaps.
Boundary values of the resolvent of a selfadjoint operator: higher order estimates
 In Algebraic and geometric methods in mathematical physics (Kaciveli
, 1993
"... Abstract. We prove in this paper resolvent estimates forthe boundary values of resolvents of selfadjoint operators on a Krein space: if H is a selfadjoint operator on a Krein space H, equipped with the Krein scalar product 〈··〉, A is the generator of a C0−group on H and I ⊂ R is an interval such th ..."
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Abstract. We prove in this paper resolvent estimates forthe boundary values of resolvents of selfadjoint operators on a Krein space: if H is a selfadjoint operator on a Krein space H, equipped with the Krein scalar product 〈··〉, A is the generator of a C0−group on H and I ⊂ R is an interval such that: 1) H admits a Borel functional calculus on I, 2) the spectral projection 1lI(H) is positive in the Krein sense, 3) the following positive commutator estimate holds: Re〈u[H,iA]u 〉 ≥ c〈uu〉, u ∈ Ran1lI(H), c> 0.
αCONTINUITY PROPERTIES OF THE SYMMETRIC αSTABLE PROCESS
, 2004
"... Abstract. Let D be a domain of finite Lebesgue measure in Rd and let XD t be the symmetric αstable process killed upon exiting D. Each element of the set {λα i}∞i=1 of eigenvalues associated to XD t, regarded as a function of α ∈ (0, 2), is right continuous. In addition, if D is Lipschitz and bound ..."
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Abstract. Let D be a domain of finite Lebesgue measure in Rd and let XD t be the symmetric αstable process killed upon exiting D. Each element of the set {λα i}∞i=1 of eigenvalues associated to XD t, regarded as a function of α ∈ (0, 2), is right continuous. In addition, if D is Lipschitz and bounded, then each λα i is continuous in α and the set of associated eigenfunctions is precompact. We also prove that if D is a domain of finite Lebesgue measure, then for all 0 < α < β ≤ 2 and i ≥ 1, λ α i ≤ λ β] α/β i Previously, this bound had been known only for β = 2 and α rational. 1.