Related papers: Direct integrals and spectral averaging
We introduce the concept of a spectral shift operator and use it to derive Krein's spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to…
In this paper, we introduce formulations of the Trotter Kato theorem for approximation of bi continuous semigroups that provide a useful framework whenever convergence of numerical approximations to solutions of PDEs are studied with…
Regularizing a linear ill-posed operator equation can be achieved by manipulating the spectrum of the operator's pseudo-inverse. Tikhonov regularization and spectral cutoff are well-known techniques within this category. This paper…
Our goal is to provide simple and practical algorithms in higher-order Fourier analysis which are based on spectral decompositions of operators. We propose a general framework for such algorithms and provide a detailed analysis of the…
In this paper spectral theorems for not necessarily continuous normal and self-adjoint random operators on a complex separable Hilbert space are proved.
The partial averaging technique is defined and used in conjunction with the random series implementation of the Feynman-Kac formula. It enjoys certain properties such as good rates of convergence and convergence for potentials with…
A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes O(m^2n)…
The universal method of expansion of integrals is suggested. It allows in particular to derive the threshold expansion of Feynman integrals.
Given two possibly unbounded selfadjoint operators A and G such that the resolvent sets of AG and GA are non-empty, it is shown that the operator AG has a spectral function on IR with singularities if there exists a non-zero polynomial p…
In this article, we prove and disprove several generalizations of unbounded versions of the Fuglede-Putnam theorem. As applications, we give conditions guaranteeing the commutativity of a bounded self-adjoint operator with an unbounded…
In this paper we analytically study the problem of pricing an arithmetically averaged Asian option in the path integral formalism. By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action…
We investigate the effect of non-symmetric relatively bounded perturbations on the spectrum of self-adjoint operators. In particular, we establish stability theorems for one or infinitely many spectral gaps along with corresponding…
It is shown that M. Krein's method of directing functionals can be used to prove the existence of a scalar spectral measure for certain Sturm-Liouville equations with two singular endpoints. The essential assumption is the existence of a…
John von Neumann's spectral theorem for self-adjoint operators is a cornerstone of quantum mechanics. Among other things, it also provides a connection between expectation values of self-adjoint operators and expected values of real-valued…
The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator valued Titchmarsh--Weyl $m$-function. This general result is applied to different self-adjoint realizations of second-order elliptic…
Inverse spectral problems are studied for the second order integro-differential operators on a finite interval. Properties of spectral characteristic are established, and the uniqueness theorem is proved for this class of inverse problems.
A direct and systematic algorithm is proposed to find one-dimensional optimal system for the group invariant solutions, which is attributed to the classification of its corresponding one-dimensional Lie algebra. Since the method is based on…
We formulate a class of singular integral operators in arbitrarily many parameters using mixed type characterizing conditions. We also prove a multi-parameter representation theorem saying that a general operator in our class can be…
This work deals with the functional model for a class of extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the…
A new approach for integration of the initial value problem for ordinary differential equations is suggested. The algorithm is based on approximation of the solution by a system of functions that contains orthogonal exponential polynomials.