543 research outputs found
The Resolvent Algebra: A New Approach to Canonical Quantum Systems
The standard C*-algebraic version of the algebra of canonical commutation
relations, the Weyl algebra, frequently causes difficulties in applications
since it neither admits the formulation of physically interesting dynamical
laws nor does it incorporate pertinent physical observables such as (bounded
functions of) the Hamiltonian. Here a novel C*-algebra of the canonical
commutation relations is presented which does not suffer from such problems. It
is based on the resolvents of the canonical operators and their algebraic
relations. The resulting C*-algebra, the resolvent algebra, is shown to have
many desirable analytic properties and the regularity structure of its
representations is surprisingly simple. Moreover, the resolvent algebra is a
convenient framework for applications to interacting and to constrained quantum
systems, as we demonstrate by several examples.Comment: 52 pages, no figures; v3: as to appear in Journal of Functional
Analysi
A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations
This paper deals with the index reduction problem for the class of
quasi-regular DAE systems. It is shown that any of these systems can be
transformed to a generically equivalent first order DAE system consisting of a
single purely algebraic (polynomial) equation plus an under-determined ODE
(that is, a semi-explicit DAE system of differentiation index 1) in as many
variables as the order of the input system. This can be done by means of a
Kronecker-type algorithm with bounded complexity
Ergodic Actions and Spectral Triples
In this article, we give a general construction of spectral triples from
certain Lie group actions on unital C*-algebras. If the group G is compact and
the action is ergodic, we actually obtain a real and finitely summable spectral
triple which satisfies the first order condition of Connes' axioms. This
provides a link between the "algebraic" existence of ergodic action and the
"analytic" finite summability property of the unbounded selfadjoint operator.
More generally, for compact G we carefully establish that our (symmetric)
unbounded operator is essentially selfadjoint. Our results are illustrated by a
host of examples - including noncommutative tori and quantum Heisenberg
manifolds.Comment: 18 page
Symmetrized Perturbation Determinants and Applications to Boundary Data Maps and Krein-Type Resolvent Formulas
The aim of this paper is twofold: On one hand we discuss an abstract approach
to symmetrized Fredholm perturbation determinants and an associated trace
formula for a pair of operators of positive-type, extending a classical trace
formula. On the other hand, we continue a recent systematic study of boundary
data maps, that is, 2 \times 2 matrix-valued Dirichlet-to-Neumann and more
generally, Robin-to-Robin maps, associated with one-dimensional Schr\"odinger
operators on a compact interval [0,R] with separated boundary conditions at 0
and R. One of the principal new results in this paper reduces an appropriately
symmetrized (Fredholm) perturbation determinant to the 2\times 2 determinant of
the underlying boundary data map. In addition, as a concrete application of the
abstract approach in the first part of this paper, we establish the trace
formula for resolvent differences of self-adjoint Schr\"odinger operators
corresponding to different (separated) boundary conditions in terms of boundary
data maps.Comment: 38 page
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