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12
Generalized solutions and spectrum for Dirichlet forms on graphs
 Progress in Probability
, 2011
"... ar ..."
On Schrödinger operators with multipolar inversesquare potentials
, 2006
"... Abstract. Positivity, essential selfadjointness, and spectral properties of a class of Schrödinger operators with multipolar inversesquare potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of po ..."
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Cited by 14 (9 self)
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Abstract. Positivity, essential selfadjointness, and spectral properties of a class of Schrödinger operators with multipolar inversesquare potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established. Contents
On Schrödinger operators with multisingular inversesquare anisotropic potentials
 Indiana Univ. Math. Journal
"... Abstract. We study positivity, localization of binding and essential selfadjointness properties of a class of Schrödinger operators with many anisotropic inverse square singularities, including the case of multiple dipole potentials. 1. Introduction and statement of the main results In this paper w ..."
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Abstract. We study positivity, localization of binding and essential selfadjointness properties of a class of Schrödinger operators with many anisotropic inverse square singularities, including the case of multiple dipole potentials. 1. Introduction and statement of the main results In this paper we analyze some basic spectral properties of Schrödinger operators associated with potentials possessing multiple anisotropic singularities of degree −2. The interest in such a class of operators arises in nonrelativistic molecular physics, where the interaction between an electric charge and the dipole moment of a molecule con be described by an inverse square potential
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 8 (6 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
, 2009
"... We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Cited by 8 (5 self)
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We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.
Exceptional circles of radial potentials
 Inverse Problems
"... Abstract. A nonlinear scattering transform is studied for the twodimensional Schrödinger equation at zero energy with a radial potential. First explicit examples are presented, both theoretically and computationally, of potentials with nontrivial singularities in the scattering transform. The sing ..."
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Cited by 4 (3 self)
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Abstract. A nonlinear scattering transform is studied for the twodimensional Schrödinger equation at zero energy with a radial potential. First explicit examples are presented, both theoretically and computationally, of potentials with nontrivial singularities in the scattering transform. The singularities arise from nonuniqueness of the complex geometric optics solutions that define the scattering transform. The values of the complex spectral parameter at which the singularities appear are called exceptional points. The singularity formation is closely related to the fact that potentials of conductivity type are “critical ” in the sense of Murata.
A2. The class Kv
"... ABSTRACT. Let H = \L + V be a general Schrödinger operator on R" (v~> 1), where A is the Laplace differential operator and V is a potential function on which we assume minimal hypotheses of growth and regularity, and in particular allow V which are unbounded below. We give a general survey ..."
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ABSTRACT. Let H = \L + V be a general Schrödinger operator on R" (v~> 1), where A is the Laplace differential operator and V is a potential function on which we assume minimal hypotheses of growth and regularity, and in particular allow V which are unbounded below. We give a general survey of the properties of e ~ tH, t> 0, and related mappings given in terms of solutions of initial value problems for the differential equation du/dt + Hu = 0. Among the subjects treated are L ^properties of these maps, existence of continuous integral kernels for them, and regularity properties of eigenfunctions, including Harnack's inequality.
QUASILINEAR YAMABE TYPE EQUATIONS WITH A SIGNCHANGING NONLINEARITY, AND THE PRESCRIBED CURVATURE PROBLEM ON NONCOMPACT MANIFOLDS
"... Abstract. In this paper, we investigate the prescribed scalar curvature problem on a noncompact Riemannian manifold (M, 〈 , 〉), namely the existence of a conformal deformation of the metric 〈 , 〉 realizing a given function s̃(x) as its scalar curvature. In particular, our work focuses on the cas ..."
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Abstract. In this paper, we investigate the prescribed scalar curvature problem on a noncompact Riemannian manifold (M, 〈 , 〉), namely the existence of a conformal deformation of the metric 〈 , 〉 realizing a given function s̃(x) as its scalar curvature. In particular, our work focuses on the case when s̃(x) changes sign, where a satisfactory theory seems to be still missing. Our main achievement is a new existence result requiring minimal assumptions on the underlying manifold, and ensuring a control on the stretching factor of the conformal deformation in such a way that the conformally deformed metric be quasiisometric to the original one. The topologicalgeometrical requirements we need are all encoded in the spectral properties of the standard and conformal Laplacians of M, which are just assumed to be subcritical (i.e. they both admit a minimal positive Green kernel on M). Another feature is that our techniques can be extended to investigate the existence of entire positive solutions of quasilinear equations of the type ∆pu+ a(x)u p−1 − b(x)uσ = 0