14 research outputs found
Topics in spectral theory of differential operators /
This dissertation is devoted to two eigenvalue counting problems: Determining the asymptotic behavior of large eigenvalues of self-adjoint extensions of partial differential operators, and computing the number of negative eigenvalues for bounded from below operators with compact resolvents. In the first part of this thesis we derive a Weyl-type asymptotic formula and a bound for the eigenvalue counting function for the Krein-von Neumann extension of differential operators on open bounded subsets of R n. In the second part of this thesis we obtain a formula relating the Maslov index, a topological invariant counting the signed number of conjugate points of paths of Lagrangian planes in H1/2 ([boundary symbol]) x H-1/2 ([boundary symbol]) and the Morse index, the number of negative eigenvalues, for the second order differential operators with domains of definition contained in H1 ([omega]) for open bounded subsets [omega] [symbol] R[subscript n].Field of study: Mathematics.|Dr. Fritz Gesztesy and Dr. Yuri Latushkin, Dissertation Supervisors.|Includes vita.Includes bibliographical references (pages 139-159)
First-order asymptotic perturbation theory for extensions of symmetric operators
This work offers a new prospective on asymptotic perturbation theory for
varying self-adjoint extensions of symmetric operators. Employing symplectic
formulation of self-adjointness we obtain a new version of Krein formula for
resolvent difference which facilitates asymptotic analysis of resolvent
operators via first order expansion for the family of Lagrangian planes
associated with perturbed operators. Specifically, we derive a Riccati-type
differential equation and the first order asymptotic expansion for resolvents
of self-adjoint extensions determined by smooth one-parameter families of
Lagrangian planes. This asymptotic perturbation theory yields a symplectic
version of the abstract Kato selection theorem and Hadamard-Rellich-type
variational formula for slopes of multiple eigenvalue curves bifurcating from
an eigenvalue of the unperturbed operator. The latter, in turn, gives a general
infinitesimal version of the celebrated formula equating the spectral flow of a
path of self-adjoint extensions and the Maslov index of the corresponding path
of Lagrangian planes. Applications are given to quantum graphs, periodic
Kronig-Penney model, elliptic second order partial differential operators with
Robin boundary conditions, and physically relevant heat equations with thermal
conductivity
A Bound for the Eigenvalue Counting Function for Higher-Order Krein Laplacians on Open Sets
For an arbitrary nonempty, open set , , of finite (Euclidean) volume, we consider the minimally defined
higher-order Laplacian , , and its Krein--von Neumann extension in
. With , , denoting the
eigenvalue counting function corresponding to the strictly positive eigenvalues
of , we derive the bound where denotes the
(Euclidean) volume of the unit ball in .
The proof relies on variational considerations and exploits the fundamental
link between the Krein--von Neumann extension and an underlying (abstract)
buckling problem.Comment: 22 pages. Considerable improvements mad
A bound for the eigenvalue counting function for Krein--von Neumann and Friedrichs extensions
For an arbitrary open, nonempty, bounded set ,
, and sufficiently smooth coefficients , we consider
the closed, strictly positive, higher-order differential operator in defined on , associated with
the higher-order differential expression and its Krein--von Neumann extension
in . Denoting by , , the eigenvalue counting function
corresponding to the strictly positive eigenvalues of , we derive the bound where (with ) is connected to the eigenfunction expansion of the self-adjoint
operator in defined on
, corresponding to . Here denotes the (Euclidean) volume of the unit ball in
.
Our method of proof relies on variational considerations exploiting the
fundamental link between the Krein--von Neumann extension and an underlying
abstract buckling problem, and on the distorted Fourier transform defined in
terms of the eigenfunction transform of in
.
We also consider the analogous bound for the eigenvalue counting function for
the Friedrichs extension in of
.
No assumptions on the boundary of are made.Comment: 39 pages. arXiv admin note: substantial text overlap with
arXiv:1403.373