Results 1  10
of
10
Perturbation of Dirichlet Forms by Measures
 POTENTIAL ANALYSIS
, 1996
"... Perturbations ofa Dirichlet form 0 by measures/ ~ are studied. The perturbed form 0 # + /z+ is defined for/~ _ in a suitable Kato class and #+ absolutely continuous with respect to capacity. Lpproperties of the corresponding semigroups are derived by approximating # _ by functions. For treating ..."
Abstract

Cited by 21 (9 self)
 Add to MetaCart
Perturbations ofa Dirichlet form 0 by measures/ ~ are studied. The perturbed form 0 # + /z+ is defined for/~ _ in a suitable Kato class and #+ absolutely continuous with respect to capacity. Lpproperties of the corresponding semigroups are derived by approximating # _ by functions. For treating #+, a criterion for domination of positive semigroups is proved. If the unperturbed semigroup has Lp Lqsmoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L ~ the same is shown to be true for the perturbed semigroup, for a large class of measures.
A convergence theorem for Dirichlet forms with applications to boundary value problems with varying domains
, 1995
"... Introduction We study continuity of boundary problems with varying domains. To explain this in more detail, let us consider our standard example: Denote by HGn the Dirichlet Laplacian on the open set G n ae IR d . The basic question which we adress is, whether we have convergence HGn \Gamma! H ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Introduction We study continuity of boundary problems with varying domains. To explain this in more detail, let us consider our standard example: Denote by HGn the Dirichlet Laplacian on the open set G n ae IR d . The basic question which we adress is, whether we have convergence HGn \Gamma! HG ; if the sets G n converge to G in an appropriate sense. Two notions of convergence for the operators appear suitable: Generalized convergence in the strong resolvent sense (srs) and in the norm resolvent sense (nrs) (the "generalized" refers to the fact that the HGn act in different Hilbert spaces; we will frequently omit it). We shall introduce these concepts in some detail below but first we briefly describe the content of the following sections. In Section 1 we a
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. ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
I.: Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
"... Abstract. 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. ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Abstract. 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.
Maximal Operators Associated With Dirichlet Forms Perturbed By Measures
 POTENTIAL ANAL
, 1998
"... It is shown that a selfadjoint operator defined by a Dirichlet form perturbed by a measure can be described as a suitable maximal operator, for a wide class of measures. The generalized `distributional notion' employed in the description reduces to the usual one for the case of Schrödinger operator ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
It is shown that a selfadjoint operator defined by a Dirichlet form perturbed by a measure can be described as a suitable maximal operator, for a wide class of measures. The generalized `distributional notion' employed in the description reduces to the usual one for the case of Schrödinger operators.
Delone measures of finite local complexity and applications to spectral theory of onedimensional continuum models of quasicrystals.DiscreteContin. Dyn.Syst
, 2011
"... Abstract. We study measures on the real line and present various versions of what it means for such a measure to take only finitely many values. We then study perturbations of the Laplacian by such measures. Using KotaniRemling theory, we show that the resulting operators have empty absolutely cont ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We study measures on the real line and present various versions of what it means for such a measure to take only finitely many values. We then study perturbations of the Laplacian by such measures. Using KotaniRemling theory, we show that the resulting operators have empty absolutely continuous spectrum if the measures are not periodic. When combined with Gordon type arguments this allows us to prove purely singular continuous spectrum for some continuum models of quasicrystals. Dedicated to Hajo Leschke on the occasion of his 65th birthday
COMPACTNESS OF SCHRÖDINGER SEMIGROUPS
"... Abstract. This paper is concerned with emptyness of the essential spectrum, or equivalently compactness of the semigroup, for perturbations of selfadjoint operators that are bounded below (on an L2space). For perturbations by a (nonnegative) potential we obtain a simple criterion for compactness of ..."
Abstract
 Add to MetaCart
Abstract. This paper is concerned with emptyness of the essential spectrum, or equivalently compactness of the semigroup, for perturbations of selfadjoint operators that are bounded below (on an L2space). For perturbations by a (nonnegative) potential we obtain a simple criterion for compactness of the semigroup in terms of relative compactness of the operators of multiplication with characteristic functions of sublevel sets. In the context of Dirichlet forms, we can even characterize compactness of the semigroup for measure perturbations. Here, certain ’averages ’ of the measure outside of compact sets play a role. As an application we obtain compactness of semigroups for Schrödinger operators with potentials whose sublevel sets are thin at infinity.
for Strongly Local Dirichlet Forms Dedicated to Jürgen Voigt in celebration of his 65th birthday
, 2008
"... Communicated by Heinz Siedentop Abstract. The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. 2000 Mathematics Subject Classification: 35P05, 81Q10 ..."
Abstract
 Add to MetaCart
Communicated by Heinz Siedentop Abstract. The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. 2000 Mathematics Subject Classification: 35P05, 81Q10
AND
"... Dedicated to Shmuel Agmon on the occasion of his 85th birthday 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. ..."
Abstract
 Add to MetaCart
Dedicated to Shmuel Agmon on the occasion of his 85th birthday 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.