1,501 research outputs found
Scattering Theory for Open Quantum Systems
Quantum systems which interact with their environment are often modeled by
maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper
the scattering theory for such open systems is considered. First it is assumed
that a single maximal dissipative operator in a Hilbert space \sH is
used to describe an open quantum system. In this case the minimal self-adjoint
dilation of can be regarded as the Hamiltonian of a closed
system which contains the open system \{A_D,\sH\}, but since
is necessarily not semibounded from below, this model is difficult to interpret
from a physical point of view. In the second part of the paper an open quantum
system is modeled with a family of maximal dissipative operators
depending on energy , and it is shown that the open system can be embedded
into a closed system where the Hamiltonian is semibounded. Surprisingly it
turns out that the corresponding scattering matrix can be completely recovered
from scattering matrices of single Pseudo-Hamiltonians as in the first part of
the paper. The general results are applied to a class of Sturm-Liouville
operators arising in dissipative and quantum transmitting
Schr\"{o}dinger-Poisson systems
On the unitary equivalence of absolutely continuous parts of self-adjoint extensions
The classical Weyl-von Neumann theorem states that for any self-adjoint
operator in a separable Hilbert space there exists a
(non-unique) Hilbert-Schmidt operator such that the perturbed
operator has purely point spectrum. We are interesting whether this
result remains valid for non-additive perturbations by considering self-adjoint
extensions of a given densely defined symmetric operator in
and fixing an extension . We show that for a wide class of
symmetric operators the absolutely continuous parts of extensions and are unitarily equivalent provided that their
resolvent difference is a compact operator. Namely, we show that this is true
whenever the Weyl function of a pair admits bounded
limits M(t) := \wlim_{y\to+0}M(t+iy) for a.e. . This result
is applied to direct sums of symmetric operators and Sturm-Liouville operators
with operator potentials
Scattering matrices and Weyl functions
For a scattering system consisting of selfadjoint
extensions and of a symmetric operator with finite
deficiency indices, the scattering matrix \{S_\gT(\gl)\} and a spectral shift
function are calculated in terms of the Weyl function associated
with the boundary triplet for and a simple proof of the Krein-Birman
formula is given. The results are applied to singular Sturm-Liouville operators
with scalar and matrix potentials, to Dirac operators and to Schr\"odinger
operators with point interactions.Comment: 39 page
Trace formulae for dissipative and coupled scattering systems
For scattering systems consisting of a (family of) maximal dissipative
extension(s) and a selfadjoint extension of a symmetric operator with finite
deficiency indices, the spectral shift function is expressed in terms of an
abstract Titchmarsh-Weyl function and a variant of the Birman-Krein formula is
proved.Comment: 38 page
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Perturbation determinants for singular perturbations
For proper extensions of a densely defined closed symmetric operator with trace class resolvent difference the perturbation determinant is studied in the framework of boundary triplet approach to extension theory
Trace formulas for singular perturbations
Trace formulas for pairs of self-adjoint, maximal dissipative and other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a resolvent comparable pair H', H of maximal dissipative operators is proved. We also investigate the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H and H'=H+V are maximal dissipative and V is of trace class, we prove the existence of a summable complex-valued spectral shift function. We also obtain trace formulas for a pair {A, A*} assuming only that A and A* are resolvent comparable. In this case the determinant of a characteristic function of A is involved in the trace formula. In the case of singular perturbations we apply the technique of boundary triplets. It allows to express the spectral shift function of a pair of extensions in terms of abstract Weyl function and boundary operator. We improve and generalize certain classical results of M.G. Krein for pairs of self-adjoint and dissipative operators, the results of A. Rybkin for such pairs, as well as the results of V. Adamyan, B. Pavlov, and M. Krein for pairs {A, A*} with a maximal dissipative operator A
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Scattering matrices and Weyl functions
For a scattering system consisting of two selfadjoint extensions of a
symmetric operator A with finite deficiency indices, the scattering matrix
and the spectral shift function are calculated in terms of the Weyl function
associated with the boundary triplet for A* and a simple proof of the
Krein-Birman formula is given. The results are applied to singular
Sturm-Liouville operators with scalar- and matrix-valued potentials, to Dirac
operators and to Schroedinger operators with point interactions
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