13,933 research outputs found
On the Exponential Stability of the Implicit Differential Systems in Hilbert Spaces
The aim of this research is to study the exponential stability of the stationary implicit system: Ax’(t) + Bx(t) = 0, where A and B are bounded operators in Hilbert spaces. The achieved results are the generalization of Liapounov Theorem for the spectrum of the operator pencil λA + B. We also establish the exponential stability conditions for the corresponding perturbed and quasi-linear implicit systems
Bounds on the spectrum and reducing subspaces of a J-self-adjoint operator
Given a self-adjoint involution J on a Hilbert space H, we consider a
J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint
operator commuting with J and V a bounded J-self-adjoint operator
anti-commuting with J. We establish optimal estimates on the position of the
spectrum of L with respect to the spectrum of A and we obtain norm bounds on
the operator angles between maximal uniformly definite reducing subspaces of
the unperturbed operator A and the perturbed operator L. All the bounds are
given in terms of the norm of V and the distances between pairs of disjoint
spectral sets associated with the operator L and/or the operator A. As an
example, the quantum harmonic oscillator under a PT-symmetric perturbation is
discussed. The sharp norm bounds obtained for the operator angles generalize
the celebrated Davis-Kahan trigonometric theorems to the case of J-self-adjoint
perturbations.Comment: (http://www.iumj.indiana.edu/IUMJ/FULLTEXT/2010/59/4225
Spectral Theory for Perturbed Krein Laplacians in Nonsmooth Domains
We study spectral properties for , the Krein--von Neumann
extension of the perturbed Laplacian defined on
, where is measurable, bounded and nonnegative, in a
bounded open set belonging to a class of nonsmooth
domains which contains all convex domains, along with all domains of class
, . In particular, in the aforementioned context we establish
the Weyl asymptotic formula #\{j\in\mathbb{N} |
\lambda_{K,\Omega,j}\leq\lambda\} = (2\pi)^{-n} v_n |\Omega|
\lambda^{n/2}+O\big(\lambda^{(n-(1/2))/2}\big) {as} \lambda\to\infty, where
denotes the volume of the unit ball in
, and , , are the non-zero
eigenvalues of , listed in increasing order according to their
multiplicities. We prove this formula by showing that the perturbed Krein
Laplacian (i.e., the Krein--von Neumann extension of defined on
) is spectrally equivalent to the buckling of a clamped
plate problem, and using an abstract result of Kozlov from the mid 1980's. Our
work builds on that of Grubb in the early 1980's, who has considered similar
issues for elliptic operators in smooth domains, and shows that the question
posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl
asymptotic formula continues to have an affirmative answer in this nonsmooth
setting.Comment: 60 page
On the point spectrum of some perturbed differential operators with periodic coefficients
Finiteness of the point spectrum of linear operators acting in a Banach space
is investigated from point of view of perturbation theory. In the first part of
the paper we present an abstract result based on analytical continuation of the
resolvent function through continuous spectrum. In the second part, the
abstract result is applied to differential operators which can be represented
as a differential operator with periodic coefficients perturbed by an arbitrary
subordinated differential operator
Quantitative bounds on the discrete spectrum of non self-adjoint quantum magnetic Hamiltonians
We establish Lieb-Thirring type inequalities for non self-adjoint relatively
compact perturbations of certain operators of mathematical physics. We apply
our results to quantum Hamiltonians of Schr{\"o}dinger and Pauli with constant
magnetic field of strength b\textgreater{}0. In particular, we use these
bounds to obtain some information on the distribution of the eigenvalues of the
perturbed operators in the neighborhood of their essential spectrum.Comment: 11 page
On elements of the Lax-Phillips scattering scheme for PT-symmetric operators
Generalized PT-symmetric operators acting an a Hilbert space
are defined and investigated. The case of PT-symmetric extensions of a
symmetric operator is investigated in detail. The possible application of
the Lax-Phillips scattering methods to the investigation of PT-symmetric
operators is illustrated by considering the case of 0-perturbed operators
A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains
In the first (and abstract) part of this survey we prove the unitary
equivalence of the inverse of the Krein--von Neumann extension (on the
orthogonal complement of its kernel) of a densely defined, closed, strictly
positive operator, for some in a Hilbert space to an abstract buckling problem operator.
This establishes the Krein extension as a natural object in elasticity theory
(in analogy to the Friedrichs extension, which found natural applications in
quantum mechanics, elasticity, etc.).
In the second, and principal part of this survey, we study spectral
properties for , the Krein--von Neumann extension of the
perturbed Laplacian (in short, the perturbed Krein Laplacian)
defined on , where is measurable, bounded and
nonnegative, in a bounded open set belonging to a
class of nonsmooth domains which contains all convex domains, along with all
domains of class , .Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144
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