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13
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
Sch’nol’s theorem for strongly local forms
, 2009
"... We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions. ..."
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Cited by 10 (6 self)
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We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ or Kirchhoff boundary conditions.
Heat kernel estimates and L^pspectral independence of elliptic operators
, 1997
"... Let\Omega be an open subset of IR d and let T p for p 2 [1; 1) be consistent C 0 semigroups given by kernels that satisfy an upper heat kernel estimate. Denoting by A p their generators, we show that the spectrum oe(A p ) is independent of p 2 [1; 1). We also treat the case of weighted L p spa ..."
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Cited by 3 (2 self)
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Let\Omega be an open subset of IR d and let T p for p 2 [1; 1) be consistent C 0 semigroups given by kernels that satisfy an upper heat kernel estimate. Denoting by A p their generators, we show that the spectrum oe(A p ) is independent of p 2 [1; 1). We also treat the case of weighted L p spaces for weights that satisfy a subexponential growth condition. An example shows that independence of the spectrum may fail for an exponential weight. We apply our result to Schrodinger operators, Petrovskij correct systems with Holder continuous coefficients, and elliptic operators in divergence form with real, but not necessarily symmetric coefficients and with complex coefficients. 1 Introduction and Main Results Let\Omega be an open subset of IR d and A be a closed linear operator in L 2(\Omega\Gamma := L 2(\Omega ; dx) where dx denotes Lebesgue measure. Assume that A generates a C 0 semigroup T in L 2(\Omega\Gamma which induces consistent C 0  semigroups T p with generators ...
THE SPECTRAL BOUND AND PRINCIPAL EIGENVALUES OF SCHRÖDINGER OPERATORS ON RIEMANNIAN MANIFOLDS
, 2001
"... Given a complete Riemannian manifold M and a Schrödinger operator − � + m acting on L p (M), we study two related problems on the spectrum of −�+m. The first one concerns the positivity of the L2spectral lower bound s(− � + m). We prove that if M satisfies L2Poincaré inequalities and a local doubl ..."
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Cited by 2 (0 self)
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Given a complete Riemannian manifold M and a Schrödinger operator − � + m acting on L p (M), we study two related problems on the spectrum of −�+m. The first one concerns the positivity of the L2spectral lower bound s(− � + m). We prove that if M satisfies L2Poincaré inequalities and a local doubling property, then s(− � + m)> 0, provided that m satisfies the mean condition 1 inf p∈M B(p, r) m(x) dx> 0
Lpanalyticity of Schrödinger semigroups on Riemannian manifolds
"... We address the problems of extrapolation, analyticity, and Lpspectral independence for C0semigroups in the abstract context of metric spaces with exponentially bounded volume. The main application of the abstract result is Lpanalyticity of angle π of Schrödinger semigroups on Riemannian manifolds ..."
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We address the problems of extrapolation, analyticity, and Lpspectral independence for C0semigroups in the abstract context of metric spaces with exponentially bounded volume. The main application of the abstract result is Lpanalyticity of angle π of Schrödinger semigroups on Riemannian manifolds with Ricci curvature 2 bounded below, under the condition of form smallness of the negative part of the potential. MSC 2000: 47D06, 47N20, 35J10 The final aim of the present paper is the following result. Let M be a Riemannian manifold with Ricci curvature bounded below. Then a Schrödinger semigroup with form small negative part of the potential is analytic of angle π 2 on Lp(M), for p from a certain subinterval of [1, ∞) that contains 2 and is determined by the form bound (see Theorem 7
On L_p spectral independence
, 2000
"... If is an open subset of R d then we prove the pindependence of the spectrum for a consistent family of bounded operators on L p( if they have a kernel K satisfying jK(x ; y)j (x y) with 2 L 1 (R d ). Extensions of this theorem are given for abstract measure spaces and for Lie groups with p ..."
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If is an open subset of R d then we prove the pindependence of the spectrum for a consistent family of bounded operators on L p( if they have a kernel K satisfying jK(x ; y)j (x y) with 2 L 1 (R d ). Extensions of this theorem are given for abstract measure spaces and for Lie groups with polynomial growth. As a result, many elliptic operators on a Lie group with polynomial growth and on a Riemannian manifold with uniform subexponentially volume growth have a pindependent spectrum. April 2000 AMS Subject Classication: 45P05, 58G25, 35P05, 47A10, 43A15. Home institutions: 1. Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands 2. Departamento de Matematica y C.C. Universidad de Santiago de Chile Casilla 307, Correo 2, Santiago Chile 1 Introduction If for all p 2 [1; 1] there is a `natural' family of operators H p in L p (X) we consider the problem whether the spectrum (H p ) of H p is independ...