Asymptotic Decoherence in Infinite-Dimensional Quantum Systems with Quadratic Hamiltonians

Mathematical Notes -     

Dynamical Systems Related to Fractional Hamiltonians on the Two-Dimensional Sphere

We consider a class of fractional Hamiltonian systems generalizing the classical problem of motion in a central field. Our analysis is based on transforming an integrable Hamiltonian system with two degrees of freedom on the plane into a dynamical system that is defined on the sphere and inherits the integrals of motion of the original system. We show that in the four-dimensional space of structural parameters, there exists a one-dimensional manifold (containing the case of the planar Kepler problem) along which the closedness of the orbits of all finite motions and the third Kepler law are preserved. Similarly, there exists a one-dimensional manifold (containing the case of the two-dimensional isotropic harmonic oscillator) along which the closedness of the orbits and the isochronism of oscillations are preserved. Any deformation of orbits on these manifolds does not violate the hidden symmetry typical of the two-dimensional isotropic oscillator and the planar Kepler problem. We also consider two-dimensional manifolds on which all systems are characterized by the same rotation number for the orbits of all finite motions.Deceased     

Deformations of Diophantine Systems for Quadratic Forms of the Cubic Lattices

The method of deformation is applied to quadratic Diophantine systems determined by the cubic lattices . The method allows one to find from known formulas for the number of representations of quadratic forms by a genus of forms an infinite set of other formulas for equations and systems with smaller number of variables. Bibliography: 5 titles.     

Irregular Isomonodromic Deformations for Garnier Systems and Okamoto's Canonical Transformations

The paper describes the Garnier systems as isomonodromic deformationequations of a linear system with a simple pole at 0 and a Poincarérank 1 singularity at infinity. The extension of Okamoto's birationalcanonical transformations to the Garnier systems in more thanone variable and to the Schlesinger systems is discussed.     

On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

We give a review of the modern theory of isomonodromic deformations of Fuchsian systems discussing both classical and modern results, such as a general form of the isomonodromic deformations of Fuchsian systems, their differences from the classical Schlesinger deformations, the Fuchsian system moduli space structure and the geometric meaning of new degrees of freedom appeared in a non-Schlesinger case. Using this we illustrate some general relations between such concepts as integrability, isomonodromy and Painlevé property. The work is supported by N.Sh.-6849.2006.1 and RFBR 07-01-00526 grants.     

具有导子的李三系的中心扩张与形变

考虑具有导子的李三系.由李三系和一个导子称为LietsDer对.定义系数在表示中的LietsDer对的上同调理论.研究LietsDer对的中心扩张.接下来,将形变理论推广到由李三系和导子构成LietsDer对上,它由带有系数的LietsDer对的上同调所支配.     

Thermodynamic Limits for Hamiltonians Defined as Pseudodifferential Operators

Abstract

If P(h) is a h-pseudodifferential operator in R ⁿ associated to an holomorphic semi-bounded symbol in some neighborhood of the real phase space, with bounded derivatives, we describe the symbol of e ^?tP(h), by inequalities where the constants depend on the bounds for the derivatives of the symbol of P(h), but not on the dimension n. Some applications to thermodynamic limits (free energy) are given.     

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

We compute the discrete affine group of Schlesinger transformations for isomonodromic deformations of a Fuchsian system of second-order differential equations. These transformations are treated as isomorphisms between the moduli spaces of logarithmic sl(2)-connections with given eigenvalues of the residues on ¹. The discrete structure is computed with the use of the modification technique for bundles with connections. The result generalizes the well-known classical computations of symmetries of the hypergeometric equation, the Heun equation, and the sixth Painlevé equation.     

The Relation and Evolution of Squeezing and Instability for Systems with Quadratic Hamiltonians

We propose a method that allows relating the quantum squeezing effect to the classical instability by establishing evolution equations for elements of the dispersion matrix directly in terms of elements of the stability matrix. The solution of these equations is written in terms of the evolution operator. Knowing this operator, we can analyze the system instability at finite times. Based on the developed formalism, we investigate two physical systems: the degenerate and nondegenerate parametric amplifiers with external -shaped pulses. We show that we can either amplify or, on the contrary, weaken both the squeezing effect and the system instability using -pulses.     

Free Vibrations of Single-branch Structural Systems

An efficient digital computer method is presented for determiningthe natural frequencies and modes of undamped free vibrationof plane single-branch structures. The structural formulationis based on use of the stiffness matrix, and the resulting eigenvalueproblem is solved by a Sturm-sequence procedure. This approachrequires considerably fewer arithmetic operations than otherstandard methods. A general computer program has been written which has provedeffective for the solution of a range of problems of practicalinterest, including the vibration of a multistorey tower, atall chimney and a long-span arched bridge. Numerical resultsare presented for the latter structure.     

Hamiltonians and conjugate Hamiltonians of some fourth-order nonlinear ODEs

We first derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov under the assumption that they satisfy the conditions stated by Fels [M.E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348, 1996, 5007-5029], using Jacobi’s last multiplier technique. In addition we derive the Hamiltonians of these equations using the Jacobi-Ostrogradski theory. Next, we derive the conjugate Hamiltonian equations for such fourth-order equations passing the Painlevé test. Finally, we investigate the conjugate Hamiltonian formulation of certain additional equations belonging to this family.     

Deformations of Diophantine Systems for Quadratic Forms of the Root Lattices A n

The paper considers a method of deformation of Diophantine quadratic systems in the n-dimensional root lattices A_n, which allows one to obtain sections of matrix quadratic equations Q[X] = A for quadratic forms Q of the lattices A_n and formulas for the number of form representations by the corresponding sections. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 5–14.     

Singly-Even Self-Dual Codes of Length 40

All singly-even self-dual [40,20,8] binary codes which have an automorphism of prime order are obtained up to equivalence. There are two inequivalent codes with an automorphism of order 7 and 37 inequivalent codes with an automorphism of order 5. These codes have highest possible minimal distance and some of them are the first known codes with weight enumerators prescribed by Conway and Sloane.     

Extensions of nonnatural Hamiltonians

Theoretical and Mathematical Physics - The concept of extended Hamiltonian systems allows a geometric interpretation of several integrable and superintegrable systems with polynomial first...     

Singularly Perturbed Hamiltonians of a Quantum Rayleigh Gas Defined as Quadratic Forms

We consider a class of singular, zero-range perturbations of the quantum Hamiltonian of a system consisting of a test particle and N harmonic oscillators (Rayleigh gas). Using the theory of quadratic forms we construct the self-adjoint and bounded from below perturbed Hamiltonian and we give representation formulas for the resolvent and the unitary group. The one-dimensional, two-dimensional and three-dimensional cases are discussed. Mathematics Subject Classifications (2000) 47A07, 47N50.Gianfausto DellAntonio: On leave from Dipartimento di Matematica, Università di Roma La Sapienza, Italy