Showing papers in "Theoretical and Mathematical Physics in 1988"
TL;DR: In this article, a Sturm-Liouville problem on compact graphs is formulated and analyzed, and the scattering problem for the Schroedinger equation on non-compact graphs is also formulated.
Abstract: A Sturm-Liouville problem on compact graphs is formulated and analyzed. The scattering problem for the Schroedinger equation on noncompact graphs is also formulated and analyzed.
147 citations
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50 citations
TL;DR: In this paper, the Miura transformation was extended to a lattice, and this made it possible to construct a completely integrable discrete variant of Liouville's equation which takes into account singular solutions.
Abstract: A proposal is made for extending the Miura transformation to a lattice, and this makes it possible to construct a completely integrable discrete variant of Liouville's equation which takes into account singular solutions
50 citations
TL;DR: In this paper, self-adjoint extensions of the operator H/sub 0/ = /minus//triangle/ from the Hilbert space L/sub 2/(R/sub n/) to a certain Pontryagin space generated by interactions represented by generalized functions are described.
Abstract: The paper describes self-adjoint extensions of the operator H/sub 0/ = /minus//triangle/ from the Hilbert space L/sub 2/(R/sub n/) to a certain Pontryagin space generated by interactions represented by generalized functions. Hamiltonians of quantum-mechanical models are obtained by restricting such extensions to positive invariant subspaces.
41 citations
TL;DR: In this article, the authors formulated a general kinematic approach for three-dimensional active media and showed that the critical curvature at the free end must be 70 cm−1, in good agreement with experimental data.
Abstract: The kinematic approach formulated in the paper is very general. The main results obtained above also remain valid for active media of a more complicated nature than the simple ones described by Eqs. (1). To construct the kinematic theory, it is necessary to know only a few phenomenological parameters such as the propagation velocity of a plane front and the critical curvature. In principle, these parameters can be calculated using the particular equations of the active medium, for example, (1). The value of the phenomenological parameters for an active medium can also be obtained experimentally. For example, for a medium with the Belousov-Zhabotinskii chemical reaction [1] the propagation velocity of a plane front is V0=2–3 mm/min, D is the diffusion coefficient for the solution and equal to D=1.8·10−5 cm2/sec, and the rotation frequency is ω=5 min−1. Then, as follows from (16), the critical curvature at the free end must be 70 cm−1, in good agreement with the experimental data. In the framework of the kinematic approach one can also consider more complicated problems of autowave propagation, for example, the behavior of spiral waves in inhomogeneous and nonstationary media. Our methods admit natural generalization to the case of three-dimensional active media.
40 citations
TL;DR: In this paper, the authors give an exposition of the combinatorial part of the proof of the generalized operator expansion at short distances in the minimal subtraction scheme based on the use of the gluing method and the counterterm technique to the case of Lagrangians and currents without normal ordering.
Abstract: The aim of this paper is to give an exposition of the combinatorial part of the proof of the generalized operator expansion at short distances in the minimal subtraction scheme based on the use of the gluing method and the counterterm technique to the case of Lagrangians and currents without normal ordering. Our approach is not based directly on expression of the renormalization procedure in terms of the action of a subtracting operator. Instead of this, we use specific features of dimensional regularization and one of the most characteristic properties of the R operation - the equivalence of this operation to the introduction of local counterterms in the Lagrangian.
38 citations
TL;DR: In this article, the massive Gross-Neveu model is treated self-consistently in the leading order of the 1/N expansion and the properties of the model when the temperature and the chemical potential are included are studied.
Abstract: The massive Gross-Neveu model is treated self-consistently in the leading order of the 1/N expansion. The properties of the model when the temperature and the chemical potential are included are studied. It is shown that there exists a critical value of the chemical potential at which the effective mass of the fermion abruptly changes its value.
TL;DR: In this paper, a detailed derivation of a covariant expression for the S matrix of an antisymmetric tensor field on the basis of the canonical formulation of the theory is given.
Abstract: We have given a detailed derivation of a covariant expression for the S matrix of an antisymmetric tensor field on the basis of the canonical formulation of the theory. An important technical point in this derivation was the introduction of an additional gauge field with a suitable transformation law. A similar trick can be used in other theories with functionally dependent constraints, in particular, as we hope, in second-quantized models of relativistic strings. Our treatment also shows that the use of the BRST quantization method necessarily requires verification of the fulfillment of all the physical conditions, since otherwise physical unitarity of the theory may be lost.
TL;DR: In this article, a study is made of small nonfinite perturbations of a Gibbs Markov random field that satisfy certain natural conditions of weakness of the correlations at sufficiently large distances.
Abstract: A study is made of small nonfinite perturbations of a Gibbs Markov random field that satisfies certain natural conditions of weakness of the correlations at sufficiently large distances. Sufficient conditions on the smallness of the perturbing potential to ensure that the analyticity of the free energy is not violated are obtained.
TL;DR: In this paper, a connection between N = 2 supersymmetric quantum mechanics and the inverse scattering problem is established, which reveals the possibility of rearrangement of the spectrum, this affecting not only the ground state but also the excited states.
Abstract: A connection between N = 2 supersymmetric quantum mechanics and the inverse scattering problem is established. In contrast to N = 1 supersymmetric quantum mechanics, construction of the isospectral Hamiltonians in the considered approach reveals the possibility of rearrangement of the spectrum, this affecting not only the ground state but also the excited states.
TL;DR: In this paper, a homogeneous and isotropic universe is studied in the framework of the relativistic theory of gravitation under the assumption that the graviton has a nonzero rest mass.
Abstract: A homogeneous and isotropic universe (Friedmann universe) is studied in the framework of the relativistic theory of gravitation under the assumption that the graviton has a nonzero rest mass. This fundamentally changes the evolution of the Friedmann universe, which becomes oscillatory; there is an infinite time and, very importantly, the matter density is always finite and nonzero.
TL;DR: In this article, the path integration method is used to describe the evolution of a quantum system subject to continuous (in time) measurement and it is shown that nonselective continuous measurement leads to a continuous increase in the degree of mixing of states.
Abstract: The path integration method is used to describe the evolution of a quantum system subject to continuous (in time) measurement. It is shown that nonselective continuous measurement leads to a continuous increase in the degree of mixing of states. A scheme is developed for calculating a family of partial evolution operators that describe the dynamics of the system with allowance for the back reaction of the instrument, and a generalized unitarity condition for them is formulated. The general results are then applied to the case of spectral measurements of a harmonic oscillator. The nature of the mixing which arises as a result of such measurements is analyzed.
TL;DR: In this article, it was shown that if certain natural conditions on the rate of decrease of the perturbing potential at arbitrarily high temperatures are violated then the free energy may cease to be analytic.
Abstract: Small nonfinite perturbations of a Gibbs random field are considered. It is shown that if certain natural conditions on the rate of decrease of the perturbing potential at arbitrarily high temperatures are violated then the free energy may cease to be analytic.
TL;DR: In this article, a recursive method is proposed for calculating the quantum corrections of arbitrary order to /var epsilon/ (/sup 0/), this being valid for both bound and quasistationary states.
Abstract: The classical approximation (/ell/ = n - 1 /yields/ /infinity/) for the energy /var epsilon/(/sup 0/) and the semiclassical expansion in problems of quantum mechanics are discussed. A recursive method is proposed for calculating the quantum corrections of arbitrary order to /var epsilon/ (/sup 0/), this being valid for both bound and quasistationary states. The generalization of the method to states with an arbitrary number of nodes and the possibility of a more general choice of the parameter of the semiclassical expansion are considered. The method is illustrated by the example of the Yukawa and funnel potentials and for the Stark effect in the hydrogen atom. These examples demonstrate the rapid convergence of the 1/n expansion even for small quantum numbers.
TL;DR: In this paper, the dynamics of a quantum-field Fermi-Bose system in the neighborhood of a nontrivial classical component in the form of an extended object of the type of a kink is considered.
Abstract: Bogolyubov's method of group variables is used to consider the dynamics of a quantum-field Fermi-Bose systems in the neighborhood of a nontrivial classical component in the form of an extended object of the type of a kink. It is shown how the quantum corrections are calculated in the framework of the method. A study is made of the kind dynamics with allowance for the quantum fluctuations that arise when it interacts with the fundamental particles of the fields; the effects due to them and related questions are discussed.