169 research outputs found
Trotter-Kato product formula for unitary groups
Let and be non-negative self-adjoint operators in a separable Hilbert
space such that its form sum is densely defined. It is shown that the
Trotter product formula holds for imaginary times in the -norm, that is,
one has % % \begin{displaymath}
\lim_{n\to+\infty}\int^T_0 \|(e^{-itA/n}e^{-itB/n})^nh - e^{-itC}h\|^2dt = 0
\end{displaymath} % % for any element of the Hilbert space and any .
The result remains true for the Trotter-Kato product formula % %
\begin{displaymath} \lim_{n\to+\infty}\int^T_0 \|(f(itA/n)g(itB/n))^nh -
e^{-itC}h\|^2dt = 0 \end{displaymath} % % where and are
so-called holomorphic Kato functions; we also derive a canonical representation
for any function of this class
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
A trace formula for functions of contractions and analytic operator Lipschitz functions
In this note we study the problem of evaluating the trace of ,
where and are contractions on Hilbert space with trace class
difference, i.e., and is a function analytic in
the unit disk . It is well known that if is an operator Lipschitz
function analytic in , then . The main
result of the note says that there exists a function (a
spectral shift function) on the unit circle of class
such that the following trace formula holds:
, whenever and are
contractions with and is an operator Lipschitz
function analytic in .Comment: 6 page
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
Remarks on the operator-norm convergence of the Trotter product formula
We revise the operator-norm convergence of the Trotter product formula for a
pair {A,B} of generators of semigroups on a Banach space. Operator-norm
convergence holds true if the dominating operator A generates a holomorphic
contraction semigroup and B is a A-infinitesimally small generator of a
contraction semigroup, in particular, if B is a bounded operator. Inspired by
studies of evolution semigroups it is shown in the present paper that the
operator-norm convergence generally fails even for bounded operators B if A is
not a holomorphic generator. Moreover, it is shown that operator norm
convergence of the Trotter product formula can be arbitrary slow.Comment: 12 page
Classical solutions of drift-diffusion equations for semiconductor devices: the 2d case
We regard drift-diffusion equations for semiconductor devices in Lebesgue
spaces. To that end we reformulate the (generalized) van Roosbroeck system as
an evolution equation for the potentials to the driving forces of the currents
of electrons and holes. This evolution equation falls into a class of
quasi-linear parabolic systems which allow unique, local in time solution in
certain Lebesgue spaces. In particular, it turns out that the divergence of the
electron and hole current is an integrable function. Hence, Gauss' theorem
applies, and gives the foundation for space discretization of the equations by
means of finite volume schemes. Moreover, the strong differentiability of the
electron and hole density in time is constitutive for the implicit time
discretization scheme. Finite volume discretization of space, and implicit time
discretization are accepted custom in engineering and scientific
computing.--This investigation puts special emphasis on non-smooth spatial
domains, mixed boundary conditions, and heterogeneous material compositions, as
required in electronic device simulation
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