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A. G. Smirnov

Publications and source records attributed to A. G. Smirnov.

At least 19 recordsLinked to original sources
arXiv2026

Measure theory without infinities

The aim of this paper is to develop a framework for measure theory that avoids infinities and allows for the uniform treatment of positive and vector measures. Our approach is based on a modification of the notion of measure, which supplements the usual $σ$-additivity requirement with a suitable maximality condition. To each Hausdorff topological vector space $\mathfrak A$ and $σ$-ring $\mathcal{Q}$, we associate a vector space $\mathscr M(\mathcal{Q},\mathfrak A)$ of `infinite' $\mathfrak A$-valued measures corresponding to $\mathcal{Q}$. In particular, the positive elements of $\mathscr M(\mathcal{Q},\mathbb R)$ are naturally identified with the $σ$-finite positive measures defined on $\mathcal{Q}$, thus placing positive and signed measures within the same setting. Finally, extension results for group-valued contents due to Sion and Weber are reformulated and refined within this new framework.

math.FA↗
arXiv2023

Completion procedures in measure theory

We propose a unified treatment of extensions of group-valued contents (i.e., additive set functions defined on a ring) by means of adding new null sets. Our approach is based on the notion of a completion ring for a content $μ$. With every such ring $\mathcal N$, an extension of $μ$ is naturally associated which is called the $\mathcal N$-completion of $μ$. The $\mathcal N$-completion operation comprises most previously known completion-type procedures and also gives rise to some new extensions, which may be useful for constructing counterexamples in measure theory. We find a condition ensuring that $σ$-additivity of a content is preserved under the $\mathcal N$-completion and establish a criterion for the $\mathcal N$-completion of a measure to be again a measure.

math.FA↗
arXiv2021

Coupling constant dependence for the Schrödinger equation with an inverse-square potential

We consider the one-dimensional Schrödinger equation $-f''+q_αf = Ef$ on the positive half-axis with the potential $q_α(r)=(α-1/4)r^{-2}$. It is known that the value $α=0$ plays a special role in this problem: all self-adjoint realizations of the formal differential expression $-\partial^2_r + q_α(r)$ for the Hamiltonian have infinitely many eigenvalues for $α<0$ and at most one eigenvalue for $α\geq 0$. We find a parametrization of self-adjoint boundary conditions and eigenfunction expansions that is analytic in $α$ and, in particular, is not singular at $α= 0$. Employing suitable singular Titchmarsh--Weyl $m$-functions, we explicitly find the spectral measures for all self-adjoint Hamiltonians and prove their smooth dependence on $α$ and the boundary condition. Using the formulas for the spectral measures, we analyse in detail how the "phase transition" through the point $α=0$ occurs for both the eigenvalues and the continuous spectrum of the Hamiltonians.

math-ph↗
arXiv2018

A Gronwall-type Trigonometric Inequality

We prove that the absolute value of the $n$th derivative of $\cos(\sqrt{x})$ does not exceed $n!/(2n)!$ for all $x>0$ and $n = 0,1,\ldots$ and obtain a natural generalization of this inequality involving the analytic continuation of $\cos(\sqrt{x})$.

math.CA↗
arXiv2016

Eigenfunction expansions for the Schrödinger equation with inverse-square potential

We consider the one-dimensional Schrödinger equation $-f"+q_κf = Ef$ on the positive half-axis with the potential $q_κ(r)=(κ^2-1/4)r^{-2}$. For each complex number $\vartheta$, we construct a solution $u^κ_\vartheta(E)$ of this equation that is analytic in $κ$ in a complex neighborhood of the interval $(-1,1)$ and, in particular, at the "singular" point $κ= 0$. For $-1<κ<1$ and real $\vartheta$, the solutions $u^κ_\vartheta(E)$ determine a unitary eigenfunction expansion operator $U_{κ,\vartheta}\colon L_2(0,\infty)\to L_2(\mathbb R,\mathcal V_{κ,\vartheta})$, where $\mathcal V_{κ,\vartheta}$ is a positive measure on $\mathbb R$. We show that every self-adjoint realization of the formal differential expression $-\partial^2_r + q_κ(r)$ for the Hamiltonian is diagonalized by the operator $U_{κ,\vartheta}$ for some $\vartheta\in\mathbb R$. Using suitable singular Titchmarsh-Weyl $m$-functions, we explicitly find the measures $\mathcal V_{κ,\vartheta}$ and prove their continuity in $κ$ and $\vartheta$.

math-ph↗
arXiv2015

Reduction by symmetries in singular quantum-mechanical problems: general scheme and application to Aharonov-Bohm model

We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general scheme is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid. We construct all self-adjoint extensions of this Hamiltonian, invariant under translations along the solenoid and rotations around it, and explicitly find their eigenfunction expansions.

math-ph↗
arXiv2013

Generators of von Neumann algebras associated with spectral measures

Let $P_E$ be the set of all values of a spectral measure $E$ and $A(P_E)$ be the smallest von Neumann algebra containing $P_E$. We give a simple description of all sets of generators of $A(P_E)$ in terms of the integrals with respect to $E$. The treatment covers not only the case of generators belonging to $A(P_E)$, but also the case of (possibly unbounded) generators affiliated with this algebra.

math.OA↗
arXiv2010

Symmetry preserving self-adjoint extensions of Schrödinger operators with singular potentials

We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general construction is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid.

math-ph↗
arXiv2008

On localization properties of Fourier transforms of hyperfunctions

In [Adv. Math. 196 (2005) 310-345] the author introduced a new generalized function space $\mathcal U(R^k)$ which can be naturally interpreted as the Fourier transform of the space of Sato's hyperfunctions on $R^k$. It was shown that all Gelfand--Shilov spaces $S^{\prime 0}_α(R^k)$ ($α>1$) of analytic functionals are canonically embedded in $\mathcal U(R^k)$. While the usual definition of support of a generalized function is inapplicable to elements of $S^{\prime 0}_α(R^k)$ and $\mathcal U(R^k)$, their localization properties can be consistently described using the concept of {\it carrier cone} introduced by Soloviev [Lett. Math. Phys. 33 (1995) 49-59; Comm. Math. Phys. 184 (1997) 579-596]. In this paper, the relation between carrier cones of elements of $S^{\prime 0}_α(R^k)$ and $\mathcal U(R^k)$ is studied. It is proved that an analytic functional $u\in S^{\prime 0}_α(R^k)$ is carried by a cone $K\subset R^k$ if and only if its canonical image in $\mathcal U(R^k)$ is carried by $K$.

math.FA↗
arXiv2007

Localization properties of highly singular generalized functions

We study the localization properties of generalized functions defined on a broad class of spaces of entire analytic test functions. This class, which includes all Gelfand--Shilov spaces $S^β_α(\R^k)$ with $β<1$, provides a convenient language for describing quantum fields with a highly singular infrared behavior. We show that the carrier cone notion, which replaces the support notion, can be correctly defined for the considered analytic functionals. In particular, we prove that each functional has a uniquely determined minimal carrier cone.

math-ph↗
arXiv2006

On kernel theorems for (LF)-spaces

A convenient technique for proving kernel theorems for (LF)-spaces (countable inductive limits of Frechet spaces)is developed. The proposed approach is based on introducing a suitable modification of the functor of the completed inductive topological tensor product. Using such modified tensor products makes it possible to prove kernel theorems without assuming the completeness of the considered (LF)-spaces. The general construction is applied to proving kernel theorems for a class of spaces of entire analytic functions arising in nonlocal quantum field theory.

math.FA↗
arXiv2005

Cohomologies of the Poisson superalgebra

Cohomology spaces of the Poisson superalgebra realized on smooth Grassmann-valued functions with compact support on $R^{2n}$ ($C^{2n}) are investigated under suitable continuity restrictions on cochains. The first and second cohomology spaces in the trivial representation and the zeroth and first cohomology spaces in the adjoint representation of the Poisson superalgebra are found for the case of a constant nondegenerate Poisson superbracket for arbitrary n>0. The third cohomology space in the trivial representation and the second cohomology space in the adjoint representation of this superalgebra are found for arbitrary n>1.

hep-th↗
arXiv2005

On kernel theorems for Frechet and DF spaces

A convenient technique for calculating completed topological tensor products of functional Frechet or DF spaces is developed. The general construction is applied to proving kernel theorems for a wide class of spaces of smooth and entire analytic functions.

math.FA↗
arXiv2004

Fourier transformation of Sato's hyperfunctions

A new generalized function space in which all Gelfand-Shilov classes $S^{\prime 0}_α$ ($α>1$) of analytic functionals are embedded is introduced. This space of {\it ultrafunctionals} does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on $R^k$ is constructed that extends the Fourier transformation of Roumieu-type ultradistributions and is naturally interpreted as the Fourier transformation of hyperfunctions. The notion of carrier cone that replaces the notion of support of a generalized function for ultrafunctionals is proposed. A Paley-Wiener-Schwartz-type theorem describing the Laplace transformation of ultrafunctionals carried by proper convex closed cones is obtained and the connection between the Laplace and Fourier transformation is established.

math.FA↗
arXiv2004

General form of deformation of Poisson superbracket

Continuous formal deformations of the Poisson superbracket defined on compactly supported smooth functions on R^n taking values in a Grassmann algebra are described up to an equivalence transformation. It is shown that there are additional deformations which are different from the standard Moyal bracket.

hep-th↗
arXiv2003

Towards Euclidean Theory of Infrared Singular Quantum Fields

A new generalized formulation of the spectral condition is proposed for quantum fields with highly singular infrared behavior whose vacuum correlation functions are well defined only under smearing with analytic test functions in momentum space. The Euclidean formulation of QFT developed by Osterwalder and Schrader is extended to theories with infrared singular indefinite metric. The corresponding generalization of the reconstruction theorem is obtained. The fulfilment of the generalized spectral condition is verified for quantum fields representable by infinite series in the Wick powers of indefinite metric free fields.

math-ph↗
arXiv2001

On Wick Power Series Convergent to Nonlocal Fields

The infinite series in Wick powers of a generalized free field are considered that are convergent under smearing with analytic test functions and realize a nonlocal extension of the Borchers equivalence classes. The nonlocal fields to which they converge are proved to be asymptotically commuting, which serves as a natural generalization of the relative locality of the Wick polynomials. The proposed proof is based on exploiting the analytic properties of the vacuum expectation values in x-space and applying the Cauchy--Poincare theorem.

math-ph↗