1,301 research outputs found
Direct and inverse spectral theorems for a class of canonical systems with two singular endpoints
Part I of this paper deals with two-dimensional canonical systems
, , whose Hamiltonian is non-negative and
locally integrable, and where Weyl's limit point case takes place at both
endpoints and . We investigate a class of such systems defined by growth
restrictions on H towards a. For example, Hamiltonians on of the
form where
are included in this class. We develop a direct and inverse spectral theory
parallel to the theory of Weyl and de Branges for systems in the limit circle
case at . Our approach proceeds via - and is bound to - Pontryagin space
theory. It relies on spectral theory and operator models in such spaces, and on
the theory of de Branges Pontryagin spaces.
The main results concerning the direct problem are: (1) showing existence of
regularized boundary values at ; (2) construction of a singular Weyl
coefficient and a scalar spectral measure; (3) construction of a Fourier
transform and computation of its action and the action of its inverse as
integral transforms. The main results for the inverse problem are: (4)
characterization of the class of measures occurring above (positive Borel
measures with power growth at ); (5) a global uniqueness theorem (if
Weyl functions or spectral measures coincide, Hamiltonians essentially
coincide); (6) a local uniqueness theorem.
In Part II of the paper the results of Part I are applied to Sturm--Liouville
equations with singular coefficients. We investigate classes of equations
without potential (in particular, equations in impedance form) and
Schr\"odinger equations, where coefficients are assumed to be singular but
subject to growth restrictions. We obtain corresponding direct and inverse
spectral theorems
Heat Kernel Asymptotics, Path Integrals and Infinite-Dimensional Determinants
We investigate the short-time expansion of the heat kernel of a Laplace type
operator on a compact Riemannian manifold and show that the lowest order term
of this expansion is given by the Fredholm determinant of the Hessian of the
energy functional on a space of finite energy paths. This is the asymptotic
behavior to be expected from formally expressing the heat kernel as a path
integral and then (again formally) using Laplace's method on the integral. We
also relate this to the zeta determinant of the Jacobi operator, which is
another way to assign a determinant to the Hessian of the energy functional. We
consider both the near-diagonal asymptotics as well as the behavior at the cut
locus.Comment: 37 pages, restructured introductio
Direct sums of trace maps and self-adjoint extensions
We give a simple criterion so that a countable infinite direct sum of trace
(evaluation) maps is a trace map. An application to the theory of self-adjoint
extensions of direct sums of symmetric operators is provided; this gives an
alternative approach to results recently obtained by Malamud-Neidhardt and
Kostenko-Malamud using regularized direct sums of boundary triplets.Comment: Final version. To appear in: S. Albeverio (ed.), Singular
Perturbation Theory, Analysis, Geometry, and Stochastic, special issue of
Arab. J. Math. (Springer
Adiabatic limits of eta and zeta functions of elliptic operators
We extend the calculus of adiabatic pseudo-differential operators to study
the adiabatic limit behavior of the eta and zeta functions of a differential
operator , constructed from an elliptic family of operators indexed by
. We show that the regularized values and
are smooth functions of at , and we identify
their values at with the holonomy of the determinant bundle, respectively
with a residue trace. For invertible families of operators, the functions
and are shown to extend smoothly to
for all values of . After normalizing with a Gamma factor, the zeta
function satisfies in the adiabatic limit an identity reminiscent of the
Riemann zeta function, while the eta function converges to the volume of the
Bismut-Freed meromorphic family of connection 1-forms.Comment: 32 pages, final versio
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