Results 1  10
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12
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 12 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Macroscopic current induced boundary conditions for Schrödingertype operators
"... We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially nonselfadjoint ..."
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Cited by 11 (9 self)
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We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially nonselfadjoint Schrödingertype operator, the spectral properties of which will be investigated.
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 7 (7 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
Eigenvalues of elliptic boundary value problems with an indefinite weight function
 TRANS. AMER. MATH. SOC
, 1986
"... ..."
Spectral Problems for the Lamé System with Spectral Parameter in Boundary Conditions on Smooth or Nonsmooth Boundary
, 1999
"... . The paper is devoted to four spectral problems for the Lame system of linear elasticity in domains of R 3 with compact connected boundary S. The frequency is xed in the upper closed halfplane; the spectral parameter enters into the boundary or transmission conditions on S. Two cases are inve ..."
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Cited by 2 (0 self)
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. The paper is devoted to four spectral problems for the Lame system of linear elasticity in domains of R 3 with compact connected boundary S. The frequency is xed in the upper closed halfplane; the spectral parameter enters into the boundary or transmission conditions on S. Two cases are investigated: (1) S is C 1 ; (2) S is Lipschitz. INTRODUCTION In this paper we consider four spectral problems for the Lame system of linear elasticity, see (1.3). The system contains the frequency parameter !, which is a xed complex number with Re! > 0. The statements of Problems I{IV are given in Subsection 1.1. The spectral parameter enters into the boundary conditions (in Problems I, II) or transmission conditions (in Problems III, IV) on a closed connected surface S, which divides its complement into a bounded domain G + and an unbounded domain G . This surface is assumed to be innitely smooth in Section 1 and Lipschitz in Section 2. Our aim is to study the spectral properties ...
Recent Developments Concerning Entropy And Approximation Numbers
 International Spring School, Nonlinear Analysis, Function Spaces and Applications V, Prague
, 1994
"... this paper,\Omega is a bounded domain with C ..."
On the existence of the Nbody E
"... In this paper, we prove the existence of the E mov eect for Nbody quantum systems with N 4. Under the conditions that the bottom of the essential spectrum, E 0 , of the Nbody operator is attained by the spectra of a unique threecluster Subhamiltonian and its three associated twocluster Subh ..."
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In this paper, we prove the existence of the E mov eect for Nbody quantum systems with N 4. Under the conditions that the bottom of the essential spectrum, E 0 , of the Nbody operator is attained by the spectra of a unique threecluster Subhamiltonian and its three associated twocluster Subhamiltonians, and that at least two of these twocluster Subhamiltonians have a resonance at the threshold E 0 , we give a lower bound of the form C 0 j log(E 0 )j for the number of eigenvalues on the left of , < E 0 , where C 0 is a positive constant depending only on the reduced masses in the threecluster decomposition. We also obtain a lower bound on the number of discrete eigenvalues in coupling constant perturbation.
Spectral Theory for . . . IN NONSMOOTH DOMAINS
, 2009
"... 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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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
Noname manuscript No.
"... (will be inserted by the editor) Greedy approximation of highdimensional Ornstein–Uhlenbeck operators with unbounded drift ..."
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(will be inserted by the editor) Greedy approximation of highdimensional Ornstein–Uhlenbeck operators with unbounded drift
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,