The Scattering Amplitude for the Schrödinger Equation with a Long-Range Potential

We consider the Schr?dinger operator with a long-range potential V(x) in the space . Our goal is to study spectral properties of the corresponding scattering matrix and a diagonal singularity of its kernel (the scattering amplitude). It turns out that in contrast to the short-range case the Dirac-function singularity of at the diagonal disappears and the spectrum of the scattering matrix covers the whole unit circle. For an asymptotically homogeneous function V(x) of order we show that typically , where the module w and the phase ψ are asymptotically homogeneous functions, as , of orders and , respectively. Leading terms of asymptotics of w and ψ at are calculated. In the case ρ=1 our results generalize (in the limit ) the well-known formula of Gordon and Mott. As a by-product of our considerations we show that the long-range scattering fits into the theory of smooth perturbations. This gives an elementary proof of existence and completeness of wave operators in the theory of long-range scattering. In this paper we concentrate on the case ρ>1/2 when the theory of pseudo-differential operators can be extensively used. Received: 29 January 1997 / Accepted: 6 May 1997     

A Change of Coordinates on the Large Phase Space¶of Quantum Cohomology

The Gromov–Witten invariants of a smooth, projective variety V, when twisted by the tautological classes on the moduli space of stable maps, give rise to a family of cohomological field theories and endow the base of the family with coordinates. We prove that the potential functions associated to the tautological ψ classes (the large phase space) and the κ classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions. Our result is a generalization of the work of Manin–Zograf who studied the case where V is a point. We utilize this change of variables to derive the topological recursion relations associated to the κ classes from those associated to the ψ classes. Received: 2 August 1999 / Accepted: 30 September 2000     

Formulation and picture of quantum phase

Based on the concept of classical phase, we formulate a new explanation for the quantum phase from the quantum mechanical point of view. The quantum phase is the canonically conjugate variable of an angular momentum operator, which corresponds to the angular position φ in an actual physical space with a classical reference frame, but it takes a complex exponential form e ^iφ≡cosφ+i sinφ in the abstract Hilbert space of a quantum reference frame. This formulation is simply the famous Euler formula in a complex number field. In particular, when φ = π/2, the correlative quantum phase is a unitary pure imaginary number e ^iπ/2≡cos(π/2)+i sin(π/2) ≡ i. By using a photon state-vector function that is the general solution of photon Schr?dinger equation and can completely describe a photon’s behavior, we discuss the relationship between the angular momentum of a photon and the phase of the photon; we also analyze the intrinsic relationship between the macroscopic light wave phase and the microscopic photon phase.     

Stochastic Dissipative PDE's and Gibbs Measures

We study a class of dissipative nonlinear PDE's forced by a random force η^{omega( t} , ^x ), with the space variable x varying in a bounded domain. The class contains the 2D Navier–Stokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in t and stationary, short-correlated in time t. In this paper, we confine ourselves to “kick forces” of the form

Phase space quantization as a moment problem

We consider questions related to the following quantization scheme: a classical variable f: Ω → ℝ on a phase space Ω is associated with a unique semispectral measure E ^f , such that the kth moment operator of E ^f is required to coincide with the operator integral L(f ^k , E) of f ^k with respect to a certain fixed phase space semispectral measure E. Mainly, we take the phase space Ω to be a locally compact unimodular group. In the concrete case where Ω = ℝ² and E is a translation covariant semispectral measure, we determine explicitly the relevant operators L(f ^k , E) for certain variables f. In addition, we consider the question under what conditions a positive operator measure is projection valued. The text was submitted by the author in English.     

Correlated coherent states in the Jaynes-Cummings model: I

In the representation of dressed states, which diagonalizes the Hamiltonian in the Jaynes-Cummings model, the Bose operator for excitation annihilation Â, the spin operator $\hat \Sigma \_In the representation of dressed states, which diagonalizes the Hamiltonian in the Jaynes-Cummings model, the Bose operator for excitation annihilation ?, the spin operator , and the corresponding Hermitian-conjugate operators are found. Using these operators, correlated coherent states were constructed and statistical properties of a field mode and an atom in these states are studied. Nonclassical field properties, such as the deviation of the probability distribution for the number of photons from the Poissonian distribution, its time dependence, and the squeezing of quadrature field components are found. For small interaction coefficients k, all these characteristics of the model vary weakly in time in the resonance case. __________ Translated from Optika i Spektroskopiya, Vol. 90, No. 6, 2001, pp. 928–934. Original Russian Text Copyright ? 2001 by Verlan, Razumova.     

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type I. The Eigenfunction Identities

In this series of papers we study Hilbert-Schmidt integral operators acting on the Hilbert spaces associated with elliptic Calogero-Moser type Hamiltonians. As shown in this first part, the integral kernels are joint eigenfunctions of differences of the latter Hamiltonians. On the relativistic (difference operator) level the kernel is built from the elliptic gamma function, whereas the building block in the nonrelativistic (differential operator) limit is basically the Weierstrass sigma-function. For the A _N−1 case we consider all of the commuting Hamiltonians at once, the eigenfunction properties reducing to a sequence of elliptic identities. For the BC _N case we only treat the defining Hamiltonians. The functional identities encoding the eigenfunction properties have a remarkable corollary in the relativistic BC ₁ case: They imply that the sum over eight-fold products of the four Jacobi theta functions is invariant under the Weyl group of E ₈.     

Differentiation of Operator Functions and Perturbation Bounds

Given a smooth real function f on the positive half line consider the induced map on the set of positive Hilbert space operators. Let f ^(k<) /E5> be the k ^th derivative of the real function f and >E5>D ^k f the k ^th Fréchet derivative of the operator map f. We identify large classes of functions for which , for k= 1,2,... . This reduction of a noncommutative problem to a commutative one makes it easy to obtain perturbation bounds for several operator maps. Our techniques serve to illustrate the use of a formalism for “quantum analysis” that is like the one recently developed by M. Suzuki. Received: 4 April 1997 / Accepted: 28 May 1997     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

Entropy increase for a class of dynamical maps

The dynamical evolution of a quantum system is described by a one parameter family of linear transformations of the space of self-adjoint trace class operators (on the Hilbert space of the system) into itself, which map statistical operators to statistical operators. We call such transformations dynamical maps. We give a sufficient condition for a dynamical map A not to decrease the entropy of a statistical operator. In the special case of an N-level system, this condition is also necessary and it is equivalent to the property that A preserves the central state.     

Nonclassical Properties of Squeezed Finite-Dimensional Pair Coherent State

We study the effect of the squeeze operator on the finite pair-coherent state. The state is a type of a correlated two-mode state in finite dimension based on the resonant ion-cavity interaction. We have discussed some statistical properties for such state, especially the quadrature variances as well as the second-order correlation function. Furthermore we have also considered the variance in the photon number sum and difference in addition to the linear correlation function. It is shown that the strong correlation occurs for a large value of the photon number difference. Our discussion is extended to include the quasiprobability distribution functions (W-Wigner and Q-functions). In the meantime we have considered the quadrature distribution function and phase distribution. It has been shown that the squeezed finite pair coherent state is sensitive to the variation in the state parameter, ξ and the squeeze parameter, r, as well as the q parameter which in fact plays a crucial role of controlling the behavior of the system.     

The Critical Renormalization Fixed Point for Commuting Pairs of Area-Preserving Maps

We prove the existence of the critical fixed point (F, G) for MacKay’s renormalization operator for pairs of maps of the plane. The maps F and G commute, are area-preserving, reversible, real analytic, and they satisfy a twist condition.     

Diffusion and relaxation of the fractional order in fractal media in the classical and quantum cases

Two model examples of the application of fractional calculus are considered. The Riemann–Liouville fractional derivative with 0 < α ≤ 1 was used. The solution of a fractional equation, which describes anomalous relaxation and diffusion in an isotropic fractal space, has been obtained in the form of the product of a Fox function by a Mittag-Leffler function. The solution is simpler than that given in Ref. 6 and it generalizes the result reported in Ref. 7. For the quantum case, a solution of the generalized Neumann–Kolmogorov fractional quantum-statistical equation has been obtained for an incomplete statistical operator which describes the random walk of a quantum spin particle, retarded in traps over a fractal space. The solution contains contributions from quantum Mittag-Leffler (nonharmonic) fractional oscillations, anomalous relaxation, noise fractional oscillations, and exponential fractional diffusion oscillation damping.     

Husimi Operator for Describing Probability Distribution of Electron States in Uniform Magnetic Field Studied by Virtue of Entangled State Representation

For the first time we introduce an operator Δ _h (γ,ε;κ) for studying Husimi distribution function in phase space (γ,ε) for electron’s states in uniform magnetic field, where κ is the Gaussian spatial width parameter. The marginal distributions of the Husimi function are Gaussian-broadened version of the Wigner marginal distributions. Using the Wigner operator in the entangled state 〈λ | representation we find that Δ _h (γ,ε;κ) is just a pure squeezed coherent state density operator | γ,ε〉 _{κ κ} 〈γ,ε |, which brings much convenience for studying Husimi distribution, so we name Δ _h (γ,ε;κ) the Husimi operator. We then derive Husimi operator’s normally ordered form that provides us with an operator version to examine various properties of the Husimi distribution. Work supported by the National Natural Science Foundation under the grant: 10775097.     

Schrödinger Operators and the Zeros of Their Eigenfunctions

In n-dimensional Euclidean space let us be given an infinitely differentiable real valued function V that is bounded below. We associate with the formal operator that sends a complex valued function ψ into −div(grad ψ) + V ψ a uniquely defined self adjoint operator which we will denote by −Δ + V.     

Transfer Operators and Dynamical Zeta Functions for a Class of Lattice Spin Models

 We investigate the location of zeros and poles of a dynamical zeta function for a family of subshifts of finite type with an interaction function depending on the parameters . The system corresponds to the well known Kac-Baker lattice spin model in statistical mechanics. Its dynamical zeta function can be expressed in terms of the Fredholm determinants of two transfer operators and with the Ruelle operator acting in a Banach space of holomorphic functions, and an integral operator introduced originally by Kac, which acts in the space with a kernel which is symmetric and positive definite for positive β. By relating via the Segal-Bargmann transform to an operator closely related to the Kac operator we can prove equality of their spectra and hence reality, respectively positivity, for the eigenvalues of the operator for real, respectively positive, β. For a restricted range of parameters we can determine the asymptotic behavior of the eigenvalues of for large positive and negative values of β and deduce from this the existence of infinitely many non-trivial zeros and poles of the dynamical zeta functions on the real β line at least for generic . For the special choice , we find a family of eigenfunctions and eigenvalues of leading to an infinite sequence of equally spaced ``trivial' zeros and poles of the zeta function on a line parallel to the imaginary β-axis. Hence there seems to hold some generalized Riemann hypothesis also for this kind of dynamical zeta functions. Received: 14 March 2002 / Accepted: 24 June 2002 Published online: 14 November 2002     

Random Analytic Chaotic Eigenstates

The statistical properties of random analytic functions (z) are investigated as a phase-space model for eigenfunctions of fully chaotic systems. We generalize to the plane and to the hyperbolic plane a theorem concerning the equidistribution of the zeros of (z) previously demonstrated for a spherical phase space [SU(2) polynomials]. For systems with time-reversal symmetry, the number of real roots is computed for the three geometries. In the semiclassical regime, the local correlation functions are shown to be universal, independent of the system considered or the geometry of phase space. In particular, the autocorrelation function of is given by a Gaussian function. The connections between this model and the Gaussian random function hypothesis as well as the random matrix theory are discussed.     

Hyper-Kähler Hierarchies and Their Twistor Theory

A twistor construction of the hierarchy associated with the hyper-K?hler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-K?hler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling–Tod (Eguchi–Hansen) solution. An extended space-time ? is constructed whose extra dimensions correspond to higher flows of the hierarchy. It is shown that ? is a moduli space of rational curves with normal bundle ?(n)⊕?(n) in twistor space and is canonically equipped with a Lax distribution for ASDVE hierarchies. The space ? is shown to be foliated by four dimensional hyper-K?hler slices. The Lagrangian, Hamiltonian and bi-Hamiltonian formulations of the ASDVE in the form of the heavenly equations are given. The symplectic form on the moduli space of solutions to heavenly equations is derived, and is shown to be compatible with the recursion operator. Received: 27 January 2000 / Accepted: 20 March 2000     

New Representation of the Self-Duality and Exact Solutions for Yang–Mills Equations

In this paper, we found a new representation for self-duality . In addition, exact solution class of the classical SU(2) Yang–Mills field in four-dimensional Euclidean space and two exact solution classes for SU(2) Yang–Mills when ρ is a complex analytic function are also obtained. PACS numbers: 11.15.-q Gauge field theories, 11.15.Kc Semiclassical theories in gauge fields, 12.10.-g, 12.15.-y Yang–Mills fields     

Natural Equilibrium States for Multimodal Maps

This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials −t log |Df|, for the largest possible interval of parameters t. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained.