Zeta-determinant and torsion functions on Riemann surfaces of finite volume

We define ζ-determinant andL ²-analytic torsion functions for a Riemann surface of finite volume. We use the Selberg trace formula to express these determinant and torsion functions in terms of four Zeta functions which are related to the structure of discrete groups. A new invariant is also obtained.     

Chiral bosonization ofU A (1)-currents and the energy-momentum tensor in quantum chromodynamics

A generalization of the chiral lagrangian is obtained by extended low-temperature bosonization of vector, axial vector, scalar, and pseudoscalar U(3) quark currents. Estimates for its structural constants are given in accordance with the developed AANN model of QCD-bosonization. Following the ideology of chiral perturbation theory, we transform the bosonized chiral Lagrangian into the phenomenological form. Relations of Zweig-rule type are established. This enables us to express the new structural constants in terms of the given constants. The correspondence between the predictions obtained by the low-temperature bosonozation method and available phenomenological estimates are discussed. Bibliography: 9 titles. In memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 55–67. Translated by V. A. Andrianov.     

Wave operators in a three-particle model of a rigid kernel

For the three-particle scattering problem in the model of a rigid kernel one establishes the existence, the isometry, and the completeness of the wave operators. One proves the absolute continuity of the Hamiltonian system.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 45–49, 1986.     

Nonaxiomatisability of equivalences over finite state processes

This paper considers the existence of finite equational axiomatisations of behavioural equivalences over a calculus of finite state processes. To express even simple properties such as μxE = μxE[E/x] some notation for substitutions is required. Accordingly, the calculus is embedded in a simply typed lambda calculus, allowing such schemas to be expressed as equations between terms containing first order variables. A notion of first order trace congruence over such terms is introduced and used to show that no finite set of such equations is sound and complete for any reasonable equivalence finer than trace equivalence. The intermediate results are then applied to give two nonaxiomatisability results over calculi of regular expressions.     

Multi-particle dynamic structure factors for a noninteracting electron gas

The three-particle dynamic structure factor for a noninteracting degenerate electron gas is calculated. A diagram technique is proposed for the dynamic structure factors of an arbitrary number of particles of a noninteracting electron gas. The symmetry properties of multi-particle dynamic structure factors are investigated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 3, pp. 438–451, March, 1996.Translated by V. I. Serdobolskii.     

Faddeev differential equations as a spectral problem for a nonsymmetric operator

We consider a nonsymmetric matrix operator whose eigenvalue problem is the system of Faddeev differential equations for a three-particle system. For this operator and its adjoint, the resolvents are represented in terms of Faddeev T-matrix components of the three-particle Schrödinger operator. On the basis of these representations, the invariant spaces of the operators under consideration are investigated and their eigenfunctions are determined. The biorthogonality and completeness of the eigenfunction system are proved.We dedicate this paper to the memory of Stanislav Petrovitch Merkuriev, who left us three years ago.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 513–528, June, 1996.     

Distribution of k-th record values in the discrete case

A result is found for discrete random variables, which lets one express the distribution of k-th record values in terms of the distribution of sums of independent summands.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 158, pp. 133–137, 1987.     

Remarks on the spectral theory for the multiparticle-type Schrödinger operator

Mourre's method is used to prove the limiting absorption principle for the multiparticle Schrödinger operator under the same assumptions on the pair potentials as in the two-particle problem. It is shown that at high energies this principle is valid under wider conditions than on the whole spectral axis. The scattering theory for a Schrödinger operator whose potential decays at infinity in an essentially anisotropic manner is constructed in analogy with the three-particle problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 277–298, 1984.     

Middle dimension homologies of a codimension 2 submanifold. II

This paper is devoted to the estimates for the middle dimension homologies of codimension 2 submanifoldA of a smooth manifold X. Proofs are based on the information about homologies of the ramified covering of X branched alongA. This information is extracted from the Atiyah-Singer signature formulas and from some corollaries of the Smith theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 182–185, 1976.The author takes this oppotunity to express his gratitude to V. A. Rokhlin for the formulation of the problem and for his interest in this paper.     

On congruences on \mathcal{T} -regular semigroups

A nontrivial regular semigroup S with zero in which every interval of idempotents is a finite chain is said to be -regular. The structure of these semigroups is described in terms of trees of completely 0-simple semigroups. For S in this form, we study congruences which we express in terms of congruence aggregates. We determine the inclusion relation, meet and join of congruences, their kernel and trace, and the ends of the intervals which form their classes. We characterize those S for which the kernel relation on the congruence lattice is a congruence, and those for which the operators and are homomorphisms.     

Spectral properties of Faddeev's equations

The spectral properties of the matrix operators corresponding to the three-particle Faddeev equations are investigated. It is shown that these operators have two types of invariant subspace. On the subspaces of the first type, the operators possess an eigenvalue spectrum identical to the spectrum of the three-particle Hamiltonian, while the eigenfunctions can be expressed in terms of solutions of the Schrödinger equation. On the subspaces of the second type, the operators are equivalent to the kinetic-energy operator of the system, and therefore their eigenfunctions do not correspond to the dynamics of the interacting particles.State University, St. Petersburg. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 323–336, March, 1995.     

Summation theorem for inductive dimensions

A bicompactum with dim = 1 and ind = Ind = 3, whichis the union of three of its own closed subsets, each of which is one-dimensional in all senses, is constructed.Translated from Matematicheskie Zametki, Vol. 52, No. 3, pp. 89–95, September, 1992.I would like to express my profound gratitude to my advisor, V. V. Filippov.     

High Energy Asymptotics and Trace Formulas for the Perturbed Harmonic Oscillator

A one-dimensional quantum harmonic oscillator perturbed by a smooth compactly supported potential is considered. For the corresponding eigenvalues λ_n, a complete asymptotic expansion for large n is obtained, and the coefficients of this expansion are expressed in terms of the heat invariants. A sequence of trace formulas is obtained, expressing regularized sums of integer powers of eigenvalues λ_n in terms of the heat invariants. Communicated by Bernard Helffer submitted 15/06/05, accepted 16/09/05     

Point interactions in the problem of three quantum particles with internal structure

A system of three quantum particles with internal structure in which the two-body interactions are point interactions and are described in terms of two-channel Hamiltonians is considered. It is established that in the cases when the parameters of the model are such that the total Hamiltonian of the three-particle system is semibounded the Faddeev equations are Fredholm equations. Boundary conditions are formulated for the differential Faddeev equations whose solutions are the scattering wave functions.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 2, pp. 258–282, February, 1995.     

Hausdorff metric structure of the space of probability measures

The Hausdorff distance between two bounded functions on the real line was introduced by B. Sendov and B. Penkov. On the class of distribution functions it agrees with the Levi distance. The goal of this article is to analyze the Levi-Prokhorov metric and a new metric in terms of the relation between their structure and the Hausdorff distance.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 87, pp. 87–103, 1979.In conclusion I would like to take this opportunity to thank V. M. Zolotarev for posing the problem and discussing the results. I express my gratitude to V. Senatov for helpful advice which helped improve the work.     

P-algebras over multidimensional local fields

This article considers simple central algebras over multidimensional local algebras of exponent p, where p is the characteristic of the zero-dimensional residue field. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 121–127, 1989.In conclusion, the author would like to express his gratitude to Professor A. S. Merkur'ev for his attention and assistance.     

Cyclic cohomology and algebra extensions

We construct a spectral sequence to study cyclic cohomology for an extension A = R/I of algebras. When R is a free algebra we describe the cyclic cohomology of A in terms of traces defined on R or powers of I. Explicit cyclic cocycles representing the cyclic cohomology class belonging to the trace are constructed as analogues of Chern character and Chern-Simons forms.Dedicated to Alexander Grothendieck     

Quasi-classical analytical approximation of theS-matrix for the three-particle collinear rearrangement reaction

In the framework of the quasi-classical method of the Fock-Leontovich parabolic equation, the collinear problem of the three-particle rearrangement reaction is reduced to the quantum problem of a one-dimensional harmonic oscillator with a variable frequency in an external force field. This allows one to express the S-matrix elements responsible for the rearrangement channel via the known analytical functions of the oscillator model, using the parameters of the potential surface of a linear configuration of three particles.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 238–250, May, 1996.Translated by T. F. Drozdova.     

The Asymptotics of the Number of Eigenvalues of a Three-Particle Lattice Schrödinger Operator

The Hamiltonian of a system of three quantum-mechanical particles moving on the three-dimensional lattice and interacting via zero-range attractive potentials is considered. The location of the essential and discrete spectra of the three-particle discrete Schrödinger operator H(K), where K is the three-particle quasimomentum, is studied. The absence of eigenvalues below the bottom of the essential spectrum of H(K) for all sufficiently small values of the zero-range attractive potentials is established.The asymptotics is found for the number of eigenvalues N(0,z) lying below . Moreover, for all sufficiently small nonzero values of the three-particle quasimomentum K, the finiteness of the number of eigenvalues below the essential spectrum of H(K) is established and the asymptotics of the number N(K,0) of eigenvalues of H(K) below zero is given.     

A supersingularity criterion of elliptic curves

A criterion for supersingularity of an elliptic curve in positive characteristic is given in terms of the mappings of this elliptic curve to a projective line that are ramified only over a single point. Bibliography: 3 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 158–160. Translated by V. V. Ishkhanov.