arXiv subjects

Sergej A. Choroszavin

Publications and source records attributed to Sergej A. Choroszavin.

14 recordsLinked to original sources

arXiv2013

Models of {\bf Z}-Orbits of Unitary in Indefinite Inner Product Spaces Operators

Given a lineal H_0 and x_0\in H_0 and a linear injective operator U_0: H_0 \to H_0 such that all U_0^N, N \in {\bf Z} exist and all U_0^N x_0, N \in {\bf Z} are linearly independent, anyone can define on span{{U_0}^N x_0 | N \in {\bf Z}} a (pre)hilbert scalar product such that U_0 becomes a unitary operator. The problem under consideration is: Suppose there is specified an indefinite inner product {,}_0 on H_0 and U_0 is a {,}_0-unitary operator. Can one introduce a (pre)hilbert topology on L_{x_0} so that after completion and possible extension the resulting {,}_ext is continuous, the resulting U_ext is {,}_ext-unitary and there exists a pair L_{+}, L_{-} mutually {,}_ext-orthogonal, maximal strictly positive and respectively negative subspaces, so that they are U_ext-invariant? More generally, can one construct a sequence (chain) of transformations of the type "restrict, change topology, make completion, extend, restrict,..." with the same result? (And, as a result, after some transformations, which are natural in the field of indefinite inner product spaces, U_ext will become usual Hilbert space unitary operator). For a relatively wide class of pairs "operator, inner product" positive solutions proposed.

math.DS↗

arXiv2013

An Example of J-unitary Operator. Solving a Problem Stated by M.G. Krein

Theorem 1. Given a number c >= 1, there exists a J-unitary operator \hat{V}, such that: (a) r(\hat{V})= r(\hat{V}^{-1})= c ; (b) S(c^{-1}\hat{V})=S(c^{-1}\hat{V}^{-1}) =S(c^{-1}\hat{V}^{*-1}) = S(c^{-1}\hat{V}^*)={0} (c) there exist maximal strictly positive and strictly negative \hat{V}^{\pm 1}-{invariant subspaces} L_{+}, L_{-}, such that they are mutually J-orthogonal and L_{+} + L_{-} is dense in the space. (d_1) if L_1 is non-zero \hat{V}-invariant subspace, then r(\hat{V}|L_1)=r(\hat{V}) (d_2) if L_2 is non-zero \hat{V}^{-1}-invariant subspace, then r(\hat{V}^{-1}|L_2)=r(\hat{V}^{-1}) .

math.DS↗

arXiv2006

What do quantum particles do, being under potential barrier? Tunnelling time. A Virtual Experiment Standpoint

Addressed, mainly: postgraduates and related readers. Subject: Given two classical mechanical 1D-moving particles (material points), with identical initial data, one of those particles given free and another given to pass through a symmetrical force-barrier, a retardation effect is observed: After the barrier has been passed over, the second particle moves with the same velocity as the free particle, but spacially is retarded with respect to the latter. If the "non-free" particle moves through a potential well, then the retarded particle is the free particle. The question is. What phenomena of a similar kind could one find, if the 1D-moving particles were quantum ones? And just what do quantum particles do, being under potential barrier? I here say "quantum" in a mathematical sense: "Schroedinger". To answer the question, I had constructed some suitable Virtual Devices (Java applets) and then for some time experimented various situations. Some of those Java applets are available at "http://choroszavin.narod.ru/vlab/index-2.htm"

math-ph↗

arXiv2004

The Ground, a String, Two Elastic Springs. Simple Exactly Solvable Models of One-Dimensional Scalar Fields with Concentrated Factors

This paper is a basis of a part of my set of lectures, subject:`Formal methods in solving differential equations and constructing models of physical phenomena'. Addressed, mainly: postgraduates and related readers. Content: a discussion of the simple models (I would rather say, toy models) of the interaction based on equation arrays of the kind: d^2 u(t,x)/dt^2=c^{2}d^2 u(t,x)/dx^2 -4{γ_a}cδ(x-x_a) Q_a(t) -4{γ_b}cδ(x-x_b) Q_b(t), Q_a(t) = u(t,x_a), Q_b(t) = u(t,x_b). Central mathematical points: d'Alembert-Kirchhoff-like Formulae, Finite Rank Perturbations.

math-ph↗

arXiv2004

Linear Hamilton Systems without Regular Properties. Solving a Problem Stated by M.G. Krein

We construct linear Hamilton systems without usual dichotomy property. The Ljapunov spectra of these systems are unfamiliar and conflicting, the behaviour of trajectories is very complicated. The paper's subject refers to some problems of indefinite inner product methods in the stability theory of abstract dynamical equation solutions.

math-ph↗

arXiv2004

Notes on Rank One Perturbed Resolvent. Perturbation of Isolated Eigenvalue

This paper is a didactic commentary (a transcription with variations) to the paper of S.R. Foguel {\it Finite Dimensional Perturbations in Banach Spaces}. Addressed, mainly: postgraduates and related readers. Subject: Suppose we have two linear operators, A, B, so that B - A is rank one. Let λ_o be an {\it isolated} point of the spectrum of A. In addition, let λ_o be an {\it eigenvalue} of A: λ_o \in σ_{pp}(A) . The question is: Is λ_o an eigenvalue of B ? And, if so, is the multiplicity of λ_o in σ_{pp}(B) equal to the multiplicity of λ_o in σ_{pp}(A) ? -- or less? -- or greater? Keywords: M.G.Krein's Formula, Finite Rank Perturbation

math-ph↗

arXiv2003

Notes on Rank-One Perturbed Resolvent

This paper is a didactic comment (a transcription with variations) to the paper of S.R. Foguel {\it Finite Dimensional Perturbations in Banach Spaces}. Addressed, mainly: postgraduates and related readers. Subject: Suppose we have two linear operators, T_1, T_2, so that T_2^{-1} - T_1^{-1} is rank-one. What we want to know is how we can express (z-T_2)^{-1} in terms of T_2^{-1} - T_1^{-1} and (z-T_1)^{-1} . Keywords: M.G.Krein's Formula, Finite Rank Perturbations.

math-ph↗

arXiv2003

A Reducing of the Invariant Semidefinite Subspace Problem for Krein Noncontraction to such a Problem for Krein Isometry

Definition. Let J be a period-2 unitary operator (some people say J is reflection operator or reflection symmetry) and U be a linear operator. If U^*JU = J (resp. U^*JU >= J) then U is said to be J-isometry (resp. J-noncontraction). If U^*JU >= J and UJU^* >= J) then U is said to be J-binoncontraction). Theorem. If every J-isometry has nontrivial positive invariant subspace then every J-noncontraction has such a subspace. Theorem. If every J-binoncontractive J-isometry has maximal positive invariant subspace then every J-noncontraction has such a subspace. The article text is the complete text of the author's report on 15-th Voronezh Winter Mathematical School, p 119 (see. VINITI 16.12.81, N 5691-81). But in that time the presented construtions and theorems seemed to be rather curious observations. Now the situattion is changing (see e.g. math.DS/9908169)

math.FA↗

arXiv2003

An Interaction of An Oscillator with An One-Dimensional Scalar Field. Simple Exactly Solvable Models based on Finite Rank Perturbations Methods. I: D'Alembert-Kirchhoff-like formulae

math.DS↗

arXiv2003

1D Particle, 1D Field, 1D Interaction. Simple Exactly Solvable Models based on Finite Rank Perturbations Methods. III. Linear Friction as Radiation Reaction

This paper is an electronic application to my set of lectures, subject:`Formal methods in solving differential equations and constructing models of physical phenomena'. Addressed, mainly: postgraduates and related readers. Content: a discussion of the simple models of linear friction, the models, that have the mechanism that is based on radiation reaction. The interactions we will deal are based on equation arrays of the kind: d^2 q(t)/dt^2 =-Ω^2 q(t)+f_{compl}(t,q,Q), d^2 u(t,x)/dt^2=c^{2}d^2 u(t,x)/dx^2 -4γcδ(x-x_0) F_{src}(t,q,Q) +f_1(t,x), Q(t) = >. Central mathematical points: d'Alembert-Kirchhoff-like formulae. Central physical points: phenomena of Radiation Reaction, Braking Radiation and Friction.

math-ph↗

arXiv2003

An Interaction of An Oscillator with An One-Dimensional Scalar Field. Simple Exactly Solvable Models based on Finite Rank Perturbations Methods. II: Resolvents formulae

This paper is an electronic application to my set of lectures, subject:`Formal methods in solving differential equations and constructing models of physical phenomena'. Addressed, mainly: postgraduates and related readers. Content: a very detailed discussion of the simple model of interaction based on the equation array: z q +Ω^2 q -Ω^2 =w_1, z u +4γ_cδ_{α,x_0}q -Bu +4γ_cδ_{α,x_0} =w_2. Besides, less detailed discussion of related models. Central mathematical points: Finite Rank Perturbations Methods, Resolvents formulae, Donoghue-like models, Friedrichs-like models. Central physical points: phenomenon of Resonance and notion of Second Sheet. Hereafter I use a P.A.M. Dirac's ``bra-ket'' syntax and suppose that $B$ stands for an abstract linear operator, $l$ for a linear functional, $u, w_2, δ_{α,x_0}$ for abstract elements; $q, w_1 z, Ω, γ_c$ stand for numbers. $q, u$ are objects to be found, the others are arbitrarily given.

math-ph↗

arXiv2003

Among Quadratic Hamiltonians, Bogoliubov Transformations and Non-Regular States on CCRs *-Algebra. I. Pure and Invariant States. (In the Mood for the Manuceau Verbeure Theorems about Quasi-free States and Automorphisma of the CCR Algebra)

The features of the paper are these: 1) we discuss {\bf especially} quadratic (alias bilinear) Bose-Hamiltonians, the related Bogoliubov transformations and {\bf especially} quasi-free-like (alias coherent or Fock-like) states 2) we discuss {\bf any} quadratic Bose-Hamiltonians and Bogoliubov transformations, whether diagonalizable or not, whether proper or improper, and {\bf arbitrary} quasi-free-like states, whether regular or non-regular they are 3) we associate notions and terms of the CCRs theory {CCRs = Canonical Commutation Relations} with notions and terms of the indefinite inner product spaces theory. Then, we apply the corresponding `bilingual dictionary' so as to construct invariant states of some of the quadratic Hamiltonians.

math-ph↗

arXiv2002

A Nonlinear Approximation of Operator Equation $V^{*}QV=Q$ : Nonspectral Decomposition of Nonnormal Operator and Theory of Stability

$V$ denotes arbitrary bounded bijection on Hilbert space $H$. We try to describe the sets of $V$-stable vectors, i.e. the set of elements $x$ of $H$ such that the sequence $\|V^N x\| (N=1,2,...)$ is bounded (we also consider some other analogous sets). We do it in terms of one-parameter operator equation $ Q_t=V^*(Q_t+tI)(I+tQ_t)^{-1}V, 0\leq Q$, ($t$ is real valued parameter $0\leq t \leq 1$,$Q$ is operator to be found $). Definition: for $t \to +0 $ denote $R_0:=w-limpt (I+Q_t)^{-1}, Y_0:= strong-lim tQ_t^{-1}, X_t:= strong-lim tQ_t $ In the case of the normal $V$ it is noted that the operators $X_0,Y_0,R_0$ define (in essential) the spectral subspaces of $V$ (with $V$ together one can consider $aV-b, b/a \not\in spectrum V$). In this article we will show that the similar situation holds for the arbitrary bounded bijection $V$.

math.DS↗

arXiv2002

Hamiltonsche Bahnen ohne Zerspaltungseigenschaft. Die Loesung einer Aufgabe von M. G. Krein

There are constructed linear Hamiltonian (dynamical) systems such that their no nonzero trajectory has usual asymptotical dichotomy property. In particular there is solved (in the negative) one of the so-called M. G. Krein problem. In fact Definition: Let J be period-2 unitary operator and U be linear operator. If U^*JU = UJU^* = J then U is said to be J-unitary. KREIN Problem: given a J-unitary operator U, does there exist an U-invariant subspace L, say, with r(U|L)\leq1 ? In the special case that the operator U^*U-I is compact this problem was solved in the positive by M.G.Krein in 1964. We shall show that, by contrast, in the general case such a subspace L needs not exist. Moreover, there asserts Theorem: For every real c>0 there exists some J-unitary operator U such that (i) if L is some nonzero U-invariant subspace, then r(U|L)>c; (ii) if L' is some nonzero U^{-1}-invariant subspace, then r(U^{-1}|L')>c; This result applies both to the real space case and to the complex space case. In addition, one can assume that U is linear symplectic automorphism. A similar result is obtained for the case of continuous 'dynamic' and for the question: does there exist a nonzero quasistable manifold?

math.DS↗