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31
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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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 17 (9 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 6 (1 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 ...
Riesz transform on locally symmetric spaces and Riemannian manifolds with a spectral gap
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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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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
Spectral properties and norm estimates associated to the C(k)c functional calculus
 J. Operator Th
"... Abstract. We show that the Davies functional calculus and the AC(ν)calculus coincide under common hypotheses. Then we apply the calculus to operators on Banach spaces, to investigate spectral invariance and norm estimates linked to abstract Cauchy equations. This extends some previous results in th ..."
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Abstract. We show that the Davies functional calculus and the AC(ν)calculus coincide under common hypotheses. Then we apply the calculus to operators on Banach spaces, to investigate spectral invariance and norm estimates linked to abstract Cauchy equations. This extends some previous results in the area, and unifies diverse approaches. The theory is applied to operators related to multiplier theory on Lie groups.
Lp spectrum and heat dynamics of locally symmetric spaces of higher rank
, 2010
"... The aim of this paper is to study the spectrum of the Lp Laplacian and the dynamics of the Lp heat semigroup on noncompact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Si ..."
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The aim of this paper is to study the spectrum of the Lp Laplacian and the dynamics of the Lp heat semigroup on noncompact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in the rank one case, it turns out that the Lp heat semigroup on M has a certain chaotic behavior if p ∈ (1, 2) whereas for p ≥ 2 such a chaotic behavior never occurs.