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Stollmann: Eigenfunction expansion for Schrödinger operators on metric graphs (Preprint arXiv:0801.1376
"... Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices. ..."
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Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.
Eigenfunction expansions for generators of Dirichlet forms
, 2003
"... We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geometry. ..."
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Cited by 5 (3 self)
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We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geometry.
Open Systems Viewed Through Their Conservative Extensions
, 2008
"... A typical linear open system is often defined as a component of a larger conservative one. For instance, a dielectric medium, defined by its frequency dependent electric permittivity and magnetic permeability is a part of a conservative system which includes the matter with all its atomic complexity ..."
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Cited by 1 (1 self)
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A typical linear open system is often defined as a component of a larger conservative one. For instance, a dielectric medium, defined by its frequency dependent electric permittivity and magnetic permeability is a part of a conservative system which includes the matter with all its atomic complexity. A finite slab of a lattice array of coupled oscillators modelling a solid is another example. Assuming that such an open system is all one wants to observe, we ask how big a part of the original conservative system (possibly very complex) is relevant to the observations, or, in other words, how big a part of it is coupled to the open system? We study here the structure of the system coupling and its coupled and decoupled components, showing, in particular, that it is only the system’s unique minimal extension that is relevant to its dynamics, and this extension often is tiny part of the original conservative system. We also give a scenario explaining why certain degrees of freedom of a solid do not contribute to its specific heat. 1 Introduction: Open