Perturbation Theory For Operators Research Articles

Application of a delta-method for random operators to testing equality of two covariance operators

In this paper some useful mathematical tools for the analysis of functional data are applied to the problem of testing the equality of two covariance operators. The test to be used is derived from a univariate likelihood ratio test in conjunction with Roy’s union-intersection principle. The asymptotic distribution and asymptotic power against rather general local alternatives are calculated. Perturbation theory of operators, in particular, a delta-method for smooth functions of operators, is exploited to obtain these theoretical results that are corroborated by some simulations.

ON LEFT AND RIGHT BROWDER OPERATORS

We discuss the perturbation theory of left and right Browder operators, which come somewhere between Browder operators and semi Browder operators.

A Perturbation Theory for Core Operators of Hilbert-Schmidt Submodules

We discuss a perturbation theory for core operators of Hilbert-Schmidt submodules in the Hardy space over the bidisk. It is shown that eigenvalues of core operators perturbed by automorphisms of the bidisk have a certain continuity, and also we show that the family of eigenspaces of simple eigenvalues has a local cross section.

Perturbation theory of lower semi-Browder multivalued linear operators

In the present paper we introduce the notion of lower semi-Browder linear relation and we study the perturbation problem under compact operator perturbations.

Operator-Lipschitz functions in Schatten–von Neumann classes

This paper resolves a number of problems in the perturbation theory of linear operators, linked with the 45-year-old conjecure of M. G. Kreĭn. In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals Sα, 1 < α < ∞. Alternatively, for every 1 < α < ∞, there is a constant cα > 0 such that $ {\left\| {f(a) - f(b)} \right\|_{\alpha }} \leqslant {c_{\alpha }}{\left\| f \right\|_{{{\text{Lip}}\,{1}}}}{\left\| {a - b} \right\|_{\alpha }}, $where f is a Lipschitz function with$ {\left\| f \right\|_{{{\text{Lip}}\,{1}}}}: = \mathop{{\sup }}\limits_{{_{{\lambda \ne \mu }}^{{\lambda, \mu \in \mathbb{R}}}}} \left| {\frac{{f\left( \lambda \right) - f\left( \mu \right)}}{{\lambda - \mu }}} \right| < \infty, $$ {\left\| \cdot \right\|_{\alpha }} $ is the norm is Sα, and a and b are self-adjoint linear operators such that $ a - b \in {S^{\alpha }} $.

Partitioning technique for open systems

The focus of the present contribution is essentially confined to three research areas carried out during the author's turns as visiting (assistant, associate and full) professor at the University of Florida's Quantum Theory Project, QTP. The first two topics relate to perturbation theory and spectral theory for self-adjoint operators in Hilbert space. The third subject concerns analytic extensions to non-self-adjoint problems, where particular consequences of the occurrence of continuous energy spectra are measured. In these studies general partitioning methods serve as general cover for perturbation-, variational- and general matrix theory. In addition we follow up associated inferences for the time dependent problem as well as recent results and conclusions of a rather general yet surprising character. Although the author spent most of his times at QTP during visits in the 1970s and 1980s, collaborations with department members and shorter stays continued through later decades. Nevertheless the impact must be somewhat fragmentary, yet it is hoped that the present account is sufficiently self-contained to be realistic and constructive.

The Dirichlet problem for second order parabolic operators in non‐cylindrical domains

AbstractIn this paper we develope a perturbation theory for second order parabolic operators in non‐divergence form. In particular we study the solvability of the Dirichlet problem in non cylindrical domains with Lp ‐data on the parabolic boundary (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Perturbation of complex polynomials and normal operators

AbstractWe study the regularity of the roots of complex monic polynomials P (t) of fixed degree depending smoothly on a real parameter t. We prove that each continuous parameterization of the roots of a generic C∞ curve P (t) (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any t0 there exists a positive integer N such that t ↦ P (t0 ± (t – t0)N ) admits smooth parameterizations of its roots near t0. We show that Cn curves P (t) (where n = deg P) admit differentiable roots if and only if the order of contact of the roots is ≥ 1. We give applications to the perturbation theory of normal matrices and unbounded normal operators with compact resolvents and common domain of definition: The eigenvalues and eigenvectors of a generic C∞ curve of such operators can be arranged locally in an absolutely continuous way (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

PENNING AND STOCHASTIC COLLISIONAL IONIZATION OF ATOMS IN AN EXTERNAL ELECTRIC FIELD

The quantum theory for the Penning and stochastic collisional ionization of atoms in an external electric field is developed and based on the operator perturbation theory and Focker-Plank stochastic equation method. Some estimates of the Penning process cross-sections for He — H, Na pairs are given.

The Bohr-Favard inequalities for operators

In this paper we estimate the resolvent of the generator of an isometric group of operators. In particular, we establish unimprovable estimates for the integral of functions that are holomorphic in a half-plane and bounded on the whole real axis. We obtain applications of the perturbation theory for linear operators.

Perturbation method for abstract second-order inverse problems

We are concerned with an abstract second-order inverse problem. We restrict ourselves here to abstract inverse problems in which, in order to solve the differential equation for u ( t ) , it is necessary to know the value of some operator or functional ϕ [ u ( t ) ] = g ( t ) as a function of the argument t . A new approach based on the perturbation theory of linear operators is introduced. Among the results obtained we point out sufficient conditions of existence and uniqueness of a solution. As a possible application of the abstract theorems, some applications from elasticity theory are given.

Analytical perturbations of wave functions and energy levels of vibrational states of molecules

A method for investigating the excited vibrational states of molecules based on the analytical perturbation theory for linear operators is proposed. According to this theory, the similarity transformation of the total projector and the construction of the transforming function allow obtaining analytical representations of eigenvectors, and in the course of reduction, eigenvalues of the perturbed operator. Theoretical statements and formulas of contributions of analytical perturbations are applied to the investigation of vibrational-rotational interactions, determination of wave functions, and corresponding energy values of excited vibrational states of molecules.

Radiative emission from triplet excitations in light emitting devices

AbstractUsing the recently derived time‐dependent spin‐orbit‐photon interaction operator and first order perturbation theory the rate of spontaneous radiative emission from triplet excitations in molecular solids and polymers is calculated within the two‐level approximation. The rate depends on the square of the atomic number of the constituting atoms. The calculated rates and corresponding radiative lifetimes agree very well with the known experimental results in conjugated polymers incorporated with platinum atoms. Present results are compared with the known results obtained from the traditional theory for naphthalene and anthracene and found in better agreement with experiments. The traditional theory is briefly reviewed. (© 2009 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

WANNIER-MOTT EXCITONS AND ATOMS IN A DC ELECTRIC FIELD: PHOTOIONIZATION, STARK EFFECT, RESONANCES IN THE IONIZATION CONTINUUM

A numerical calculation of the DC Stark effect for hydrogen and sodium atoms and Wannier-Mott excitons in an external uniform DC electric field is carried out within the operator perturbation theory method. New adequate model for description of the unique especialitites in the photoionization spectra of the hydrogenic atoms is proposed. It is found that the Stark shift for the n=2 state of excitons in the Cu2 O semiconductor (yellow series) at the electric field strength 600 V/cm results in — 3,1 10 -4 eV which agrees well with experimental data of Gross et al. It is indicated also that the analogous unique especialitites may possibly take a place in the Wannier-Mott excitons spectra near the threshold boundary

Stochastic uniform observability of general linear differential equations

The aim of this article is to discuss the uniform observability property of general linear differential equations with multiplicative white noise in Hilbert spaces. Based on perturbation theory for evolution operators on Hilbert Schmidt spaces and on the space of nuclear operators, new representations of the covariance operators associated to the mild solutions of the investigated stochastic differential equations are given. Using these results we obtain deterministic characterizations of the stochastic uniform observability property. We also identify an entire class of stochastic differential equations which are never stochastic uniformly observable. Some examples will illustrate the theory.

A direct approach to study radiative emission from triplet excitations in molecular semiconductors and conjugated polymers

Using the recently discovered time-dependent spin-orbit-photon interaction operator and first order perturbation theory, the rate of spontaneous emission from triplet excitations is derived within the two-level approximation for organic molecular solids and conjugated polymers. The calculated rates and corresponding radiative lifetimes agree very well with the known experimental results. Present results are compared with those obtained through the traditional approach of the second order perturbation theory in some molecular crystals and found to be in better agreement with experiments.

Spectral methods for orthogonal rational functions

We present an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two alternatives are discussed, leading to representations which are linear fractional transformations with matrix coefficients acting on infinite Hessenberg or five-diagonal unitary matrices. This approach permits us to recover the orthogonality measure throughout the spectral analysis of an infinite matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg or five-diagonal matrices. As an application we use operator perturbation theory results to obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation for the orthogonal rational functions.

Perturbation theory of multivalued atkinson operators in normed spaces

We prove several stability results for Atkinson linear relations under additive perturbation by small norm, strictly singular and strictly cosingular multivalued linear operators satisfying some additional conditions.

Computing the sums of Rayleigh-Schrödinger series of perturbed self-adjoint operators

A new method is proposed and validated for computing the sums of number series of discrete operators in perturbation theory with the required accuracy. Its efficiency is demonstrated through numerical computations.

Strong-coupling limit of the Richardson Hamiltonian analyzed using Dyson mapping

The Richardson Hamiltonian describes superconducting correlations in a metallic nanograin. We do a perturbative analysis of this and related Hamiltonians, around the strong pairing limit, without having to invoke Bethe Ansatz solvability. Rather we make use of a boson expansion method known as the Dyson mapping. Thus we uncover a selection rule that facilitates both time-independent and time-dependent perturbation expansions. In principle the model we analise is realised in a very small metalic grain of a very regular shape. The results we obtain point to subtleties sometimes neglected when thinking of the superconducting state as a Bose-Einstein condensate. An appendix contains a general presentation of time-independent perturbation theory for operators with degenerate spectra, with recursive formulas for corrections of arbitrarily high orders.