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13
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.
NOTE ON BASIC FEATURES OF LARGE TIME BEHAVIOUR OF HEAT KERNELS
"... Abstract. Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoi ..."
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Cited by 11 (8 self)
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Abstract. Large time behaviour of heat semigroups (and more generally, of positive selfadjoint semigroups) is studied. Convergence of the semigroup to the ground state and of averaged logarithms of kernels to the ground state energy is shown in the general framework of positivity improving selfadjoint semigroups. This framework includes Laplacians on manifolds, metric graphs and discrete graphs.
Anderson localization for radial treelike random quantum graphs
, 2008
"... We prove that certain random models associated with radial, treelike, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family ..."
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Cited by 11 (0 self)
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We prove that certain random models associated with radial, treelike, rooted quantum graphs exhibit Anderson localization at all energies. The two main examples are the random length model (RLM) and the random Kirchhoff model (RKM). In the RLM, the lengths of each generation of edges form a family of independent, identically distributed random variables (iid). For the RKM, the iid random variables are associated with each generation of vertices and moderate the current flow through the vertex. We consider extensions to various families of decorated graphs and prove stability of localization with respect to decoration. In particular, we prove Anderson localization for the random necklace model.
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.
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 ..."
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Cited by 4 (1 self)
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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.
New Relations Between Discrete and Continuous Transition Operators on (Metric) Graphs
"... We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators can be expressed through each other using explicit expressions. In particular, we show ..."
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We establish several new relations between the discrete transition operator, the continuous Laplacian and the averaging operator associated with combinatorial and metric graphs. It is shown that these operators can be expressed through each other using explicit expressions. In particular, we show that the averaging operator is closely related with the solutions of the associated wave equation. The machinery used allows one to study a class of infinite graphs without assumption on the local finiteness. 1
SCH’NOL’S THEOREM FOR STRONGLY LOCAL FORMS
, 708
"... Abstract. 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. Dedicated to Shmuel Agmon on the occasion of his 85th birthday ..."
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Abstract. 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. Dedicated to Shmuel Agmon on the occasion of his 85th birthday