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On Inverse Spectral Theory for SelfAdjoint Extensions: Mixed Types of Spectra
 J. Funct. Anal
, 1996
"... Let H be a symmetric operator in a separable Hilbert space H. Suppose that H has some gap J . We shall investigate the question about what spectral properties the selfadjoint extensions of H can have inside the gap J and provide methods how to construct selfadjoint extensions of H with prescrib ..."
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Cited by 10 (3 self)
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Let H be a symmetric operator in a separable Hilbert space H. Suppose that H has some gap J . We shall investigate the question about what spectral properties the selfadjoint extensions of H can have inside the gap J and provide methods how to construct selfadjoint extensions of H with prescribed spectral properties inside J . Under some weak assumptions about the operator H which are satisfied, e. g., provided the deficiency indices of H are infinite and the operator (H \Gamma ) \Gamma1 is compact for one regular point of H, we shall show that for every (auxiliary) selfadjoint operator M 0 in the Hilbert space H and every open subset J 0 of the gap J of H there exists a selfadjoint extension ~ H of H such that inside J the selfadjoint extension ~ H of H has the same absolutely continuous and the same point spectrum as the given operator M 0 and the singular continuous spectrum of ~ H in J equals the closure of J 0 in J . Moreover we shall present a method how to ...
On a relation between intrinsic and extrinsic Dirichlet forms for interacting particle systems
, 1998
"... In this paper we extend the result obtained in [AKR98] (see also [AKR96a]) on the representation of the intrinsic preDirichlet form E \Gamma ß oe of the Poisson measure ß oe in terms of the extrinsic one E \Gamma ß oe ;H X oe . More precisely, replacing ß oe by a Gibbs measure ¯ on the confi ..."
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Cited by 6 (0 self)
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In this paper we extend the result obtained in [AKR98] (see also [AKR96a]) on the representation of the intrinsic preDirichlet form E \Gamma ß oe of the Poisson measure ß oe in terms of the extrinsic one E \Gamma ß oe ;H X oe . More precisely, replacing ß oe by a Gibbs measure ¯ on the configuration space \Gamma X we derive a relation between the intrinsic preDirichlet form E \Gamma ¯ of the measure ¯ and the extrinsic one E P ¯;H X oe . As a consequence we prove the closability of E \Gamma ¯ on L 2 (\Gamma X ; ¯) under very general assumptions on the interaction potential of the Gibbs measures µ.
Schrödinger Operator Perturbed By Operators Related to Null Sets
"... We discuss the Schrödinger operator with singular perturbations given by operators which act in the space constructed by a measure supported byanull set. We construct examples when perturbations are given by the onedimensional Laplacian on a segment. ..."
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Cited by 4 (0 self)
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We discuss the Schrödinger operator with singular perturbations given by operators which act in the space constructed by a measure supported byanull set. We construct examples when perturbations are given by the onedimensional Laplacian on a segment.
A general correspondence between Dirichlet forms and right processes
 Bull. Amer. Math. Soc
, 1991
"... The theory of Dirichlet forms as originated by BeurlingDeny and developed particularly by Fukushima and Silverstein, see e.g. [Fu3, Si], is a natural functional analytic extension of classical (and axiomatic) potential theory. Although some parts of it have abstract measure theoretic versions, see ..."
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Cited by 2 (0 self)
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The theory of Dirichlet forms as originated by BeurlingDeny and developed particularly by Fukushima and Silverstein, see e.g. [Fu3, Si], is a natural functional analytic extension of classical (and axiomatic) potential theory. Although some parts of it have abstract measure theoretic versions, see e.g. [BoH] and [ABrR],
An Extended Generator and Schrödinger Equations
 ELEC. J. PROBAB
, 1999
"... The generator of a Borel right process is extended so that it maps functions to smooth measures. This extension may be defined either probabilistically using martingales or analytically in terms of certain kernels on the state space of the process. Then the associated Schrodinger equation with a (si ..."
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Cited by 1 (0 self)
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The generator of a Borel right process is extended so that it maps functions to smooth measures. This extension may be defined either probabilistically using martingales or analytically in terms of certain kernels on the state space of the process. Then the associated Schrodinger equation with a (signed) measure serving as potential may be interpreted as an equation between measures. In this context general existence and uniqueness theorems for solutions are established. These are then specialized to obtain more concrete results in special situations.
On Singular Perturbations of Order S, S <= 2, of the Free Dynamics: Existence And Completeness of . . .
, 2002
"... For a large class of operators H a formula for the resolvent will be derived and sufficient conditions will be given in order that the resolvent difference (H \Gamma z) is compact resp. the wave (H; H 0 ) exist and are complete; here H 0 denotes the free quantum mechanical Hamiltonian. The ..."
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For a large class of operators H a formula for the resolvent will be derived and sufficient conditions will be given in order that the resolvent difference (H \Gamma z) is compact resp. the wave (H; H 0 ) exist and are complete; here H 0 denotes the free quantum mechanical Hamiltonian. The mentioned class of operators contains, a.o., the generator of a Brownian motion with killing, the generator of the superposition of a Brownian motion and a diffusion process on a submanifold and Hamiltonians describing the interaction of a quantum mechanical particle with a potential supported by a set with classical capacity zero.
Schrödinger Operator Perturbed By Operators Related to Null Sets
"... We discuss the Schrödinger operator with singular perturbations given by operators which act in the space constructed by a measure supported byanull set. We construct examples when perturbations are given by the onedimensional Laplacian on a segment. ..."
Abstract
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We discuss the Schrödinger operator with singular perturbations given by operators which act in the space constructed by a measure supported byanull set. We construct examples when perturbations are given by the onedimensional Laplacian on a segment.