On the exponential decay of correlation functions

We use the properties of subharmonic functions to prove the following results, First, for any lattice system with finite-range forces there is a gap in the spectrum of the transfer matrix, which persists in the thermodynamic limit, if the fugacityz lies in a regionE of the complex plane that contains the origin and is free of zeros of the grand partition function (with periodic boundary conditions) as the thermodynamic limit is approached. Secondly, if the transfer matrix is symmetric (for example, with nearest and next-nearest neighbor interactions in two dimensions), and if infinite-volume Ursell functions exist that are independent of the order in which the various sides of the periodicity box tend to infinity, then these Ursell functions decay exponentially with distance for all positivez inE. (For the Ising ferromagnet with two-body interactions, exponential decay holds forz inE even if the range of interaction is not restricted to one lattice spacing). Thirdly, if the interaction potential decays moreslowly than any decaying exponential, then so do all the infinite-volume Ursell functions, for almost all sufficiently small fugacities in the case of general lattice systems, and for all real magnetic fields in the case of Ising ferromagnets.     

On the Discrete Spectrum of a Spatial Quantum Waveguide with a Disc Window

In this study we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width d. We impose the Neumann boundary condition on a disc window of radius a and Dirichlet boundary conditions on the remained part of the boundary of the strip. We prove that such system exhibits discrete eigenvalues below the essential spectrum for any a > 0. We give also a numeric estimation of the number of discrete eigenvalue as a function of \fracad\frac{a}{d}. When a tends to the infinity, the asymptotic of the eigenvalue is given.     

Root Asymptotics of Spectral Polynomials for the Lamé Operator

The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schrödinger equation with finite gap potential given by the Weierstrass $\wpThe study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schr?dinger equation with finite gap potential given by the Weierstrass -function on the real line. In this paper we establish several natural (and equivalent) formulas in terms of hypergeometric and elliptic type integrals for the density of the appropriately scaled asymptotic distribution of these eigenvalues when the integer-valued spectral parameter tends to infinity. We also show that this density satisfies a Heun differential equation with four singularities.     

On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices

We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U- and V- (von Mises) statistics of eigenvalues of random matrices as their size tends to infinity. We show first that for a certain class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for several most known ensembles of random matrices. The reduction phenomenon is well known in statistics, dealing with i.i.d. random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are strongly dependent even if the entries of matrices are independent.     

On the Born-Oppenheimer expansion for polyatomic molecules

We consider the Schrödinger operatorP(h) for a polyatomic molecule in the semiclassical limit where the mass ratioh ² of electronic to nuclear mass tends to zero. We obtain WKB-type expansions of eigenvalues and eigenfunctions ofP(h) to all orders inh. This allows to treat the splitting of the ground state energy of a non-planar molecule. Our class of potentials covers the physical case of the Coulomb interaction. We use methods ofh-pseudodifferential operators with operator valued symbols, which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials.     

Density of Complex Zeros of a System of Real Random Polynomials

We study the density of complex zeros of a system of real random SO(m+1) polynomials in m variables. We show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case. We also show that the behavior the scaled density of complex zeros near ℝ ^m of the system of real random polynomials is different in the m≥2 case than in the m=1 case: the density approaches infinity instead of tending linearly to zero.     

A particle in a box in the presence of an electric field and applications to disordered systems

The Green's function and energy eigenvalues of an electron under the influence of a uniform electric field in a box with infinitely high sides is investigated. The second order correction to the energy eigenvalues is calculated by finding zeros of the wronskian and comparing with the value obtained from second order perturbation theory. Comparison is made with the limiting conditions in which the size of the box tends to infinity and the electric field tends towards zero. The results of the investigation suggest a possible criterion for localisation. The value obtained for the ground state energy is used to extend a model of Edwards to study the tail of the density of states of a disordered system in the presence of an electric field.     

On the Crystallization of 2D Hexagonal Lattices

It is a fundamental problem to understand why solids form crystals at zero temperature and how atomic interaction determines the particular crystal structure that a material selects. In this paper we focus on the zero temperature case and consider a class of atomic potentials V = V ₂ + V ₃, where V ₂ is a pair potential of Lennard-Jones type and V ₃ is a three-body potential of Stillinger-Weber type. For this class of potentials we prove that the ground state energy per particle converges to a finite value as the number of particles tends to infinity. This value is given by the corresponding value for a optimal hexagonal lattice, optimized with respect to the lattice spacing. Furthermore, under suitable periodic or Dirichlet boundary condition, we show that the minimizers do form a hexagonal lattice. Dedicated with admiration to Professor Tom Spencer on occasion of his 60th birthday     

On the Existence of the Absolutely Continuous Component for the Measure Associated¶with Some Orthogonal Systems

In this article, we consider two orthogonal systems: Sturm–Liouville operators and Krein systems. For Krein systems, we study the behavior of generalized polynomials at the infinity for spectral parameters in the upper half-plain. That makes it possible to establish the presence of absolutely continuous component of the associated measure. For Sturm–Liouville operator on the half-line with bounded potential q, we prove that essential support of absolutely continuous component of the spectral measure is [m,∞) if and q ^′∈L ²(R ⁺). That holds for all boundary conditions at zero. This result partially solves one open problem stated recently by S. Molchanov, M. Novitskii, and B. Vainberg. We consider also some other classes of potentials. Received: 26 June 2001 / Accepted: 18 October 2001     

On Asymptotic Expansions and Scales of Spectral Universality in Band Random Matrix Ensembles

 We consider real random symmetric N × N matrices H of the band-type form with characteristic length b. The matrix entries are independent Gaussian random variables and have the variance proportional to , where u(t) vanishes at infinity. We study the resolvent in the limit and obtain the explicit expression for the leading term of the first correlation function of the normalized trace . We examine on the local scale and show that its asymptotic behavior is determined by the rate of decay of u(t). In particular, if u(t) decays exponentially, then . This expression is universal in the sense that the particular form of u determines the value of C > 0 only. Our results agree with those detected in both numerical and theoretical physics studies of spectra of band random matrices. Received: 8 April 2000 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Present address: Département de Mathématiques, Université de Versailles Saint-Quentin, 78035 Versailles, France.     

Neutron matter with a model interaction

An infinite system of neutrons interacting by a model pair potential is considered. We investigate a case when this potential is sufficiently strong attractive, so that its scattering length a tends to infinity, a . It appeared, that if the structure of the potential is simple enough, including no finite parameters, reliable evidences can be presented that such a system is completely unstable at any finite density. The incompressibility as a function of the density is negative, reaching zero value when the density tends to zero. If the potential contains a sufficiently strong repulsive core then the system possesses an equilibrium density. The main features of a theory describing such systems are considered. Received: 17 December 1999 / Accepted: 23 December 1999     

Application of asymptotic iteration method to the Khare-Mandal and its PT-symmetric QES partner potential

We study the application of the asymptotic iteration method to the Khare-Mandal potential and its PT-symmetric partner. The eigenvalues and eigenfunctions for both potentials are obtained analytically. We have shown that although the quasi-exactly solvable energy eigenvalues of the Khare-Mandal potential are found to be in complex conjugate pairs for certain values of potential parameters, its PT-symmetric partner exhibits real energy eigenvalues in all cases.      

Perturbation theory for velocity-dependent potentials

In the presence of a velocity-dependent Kisslinger potential, the partial-wave, time-independent Schr?dinger equation with real boundary conditions is written as an equation for the probability density. The changes in the bound-state energy eigenvalues due to the addition of small perturbations in the local as well as the Kisslinger potentials are determined up to second order in the perturbation. These changes are determined purely in terms of the unperturbed probability density, the perturbing local potential, as well as the Kisslinger perturbing potential and its gradient. The dependence on the gradient of the Kisslinger potential stresses the importance of a diffuse edge in nuclei. Two explicit examples are presented to examine the validity of the perturbation formulas. The first assumes each of the local and velocity-dependent parts of the potential to be a finite square well. In the second example, the velocity-dependent potential takes the form of a harmonic oscillator. In both cases the energy eigenvalues are determined exactly and then by using perturbation theory. The agreement between the exact energy eigenvalues and those obtained by perturbation theory is very satisfactory. Received: 24 May 2002 / Accepted: 15 July 2002 / Published online: 3 December 2002 RID="a" ID="a"e-mail: mij@hu.edu.jo Communicated by V. Vento     

Eigenvalue Boundary Problems for the Dirac Operator

 On a compact Riemannian spin manifold with mean-convex boundary, we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental Dirac operator including the (global) APS condition and a Riemannian version of the (local) MIT bag condition. We show that Friedrich's inequality for the eigenvalues of the Dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems. More precisely, we prove that, for both the APS and the MIT conditions, the equality cannot be achieved, and for the other two conditions, the equality characterizes respectively half-spheres and domains bounded by minimal hypersurfaces in manifolds carrying non-trivial real Killing spinors. Received: 12 November 2001 / Accepted: 25 June 2002 Published online: 21 October 2002 RID="*" ID="*" Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001-2967 and by European Union FEDER funds     

A new (original) set of Quasi-normal modes in spherically symmetric AdS black hole spacetimes

From black hole perturbation theory, quasi-normal modes (QNMs) in spherically symmetric AdS black hole spacetimes are usually studied with the Horowitz and Hubeny methods [1] by imposing the Dirichlet or vanishing energy flux boundary conditions. This method was constructed using the scalar perturbation case and box-like effective potentials, where the radial equation tends to go to infinity when the radial coordinate approaches infinity. These QNMs can be realized as a different set of solutions from those obtained by the barrier-like effective potentials. However, in some cases the existence of barrier-like effective potentials in AdS black hole spacetimes can be found. In these cases this means that we would obtain a new (original) set of QNMs by the purely ingoing and purely outgoing boundary conditions when the radial coordinate goes to the event horizon and infinity, respectively. Obtaining this set of QNMs in AdS black hole cases is the main focus of this paper.     

Semi-Dispersing Billiards with an Infinite Cusp I

 Let be a sufficiently smooth convex function, vanishing at infinity. Consider the planar domain Q delimited by the positive x-semiaxis, the positive y-semiaxis, and the graph of f. Under certain conditions on f, we prove that the billiard flow in Q has a hyperbolic structure and, for some examples, that it is also ergodic. This is done using the cross section corresponding to collisions with the dispersing part of the boundary. The relevant invariant measure for this Poincaré section is infinite, whence the need to surpass the existing results, designed for finite-measure dynamical systems. Received: 1 May 2002 / Accepted: 13 May 2002 Published online: 22 August 2002     

The Cosmic Time in Terms of the Redshift

In cosmology one labels the time t since the Big Bang in terms of the redshift of light emitted at t, as we see it now. In this Note we derive a formula that relates t to z which is valid for all redshifts. One can go back in time as far as one wishes, but not to the Big Bang at which the redshift tends to infinity.     

Kink stability of isothermal spherical self-similar flow revisited

The problem of kink stability of isothermal spherical self-similar flow in newtonian gravity is revisited. Using distribution theory we first develop a general formula of perturbations, linear or non-linear, which consists of three sets of differential equations, one in each side of the sonic line and the other along it. By solving the equations along the sonic line we find explicitly the spectrum, k, of the perturbations, whereby we obtain the stability criterion for the self-similar solutions. When the solutions are smoothly across the sonic line, our results reduce to those of Ori and Piran. To show such obtained perturbations can be matched to the ones in the regions outside the sonic line, we study the linear perturbations in the external region of the sonic line (the ones in the internal region are identically zero), by taking the solutions obtained along the line as the boundary conditions. After properly imposing other boundary conditions at spatial infinity, we are able to show that linear perturbations, satisfying all the boundary conditions, exist and do not impose any additional conditions on k. As a result, the complete treatment of perturbations in the whole spacetime does not alter the spectrum obtained by considering the perturbations only along the sonic line.     

Multiple scattering theory in condensed materials

Second-order elliptic differential equations (such as the time-independent single particle Schrödinger equation) may be solved in a finite closed disjoint region of space independently of the rest of space. The solution in all space may then be determined by solving the equations in the exterior region together with boundary conditions at the junction of the two regions. These boundary conditions are determined by the previously found interior solution. This means that such regions may be taken as ‘black boxes’ whose exact details do not matter. The simplest example of this is phase-shift scattering theory from a single scatterer where all the scattering properties are described by the phase shifts, and the exact details of the scattering potential are unimportant. In a macroscopic condensed system, however, there are many core regions and one is really concerned with the multiple scattering which takes place between these different scattering centres. Much of this article is devoted to investigating the formal properties of scattering theory when there are many non-overlapping spherical regions of radius R _M, each of which is described by its own scattering matrix, or, equivalently for a spherically symmetric potential, by its phase shifts. Non-spherically symmetric and spin-dependent potentials are permitted, but for simplicity we assume initially that the interstitial region between each disjoint scattering region has zero potential. The generalization of the multiple scattering formalism for non-zero interstitial potential is also given at a later stage.

It is shown that in such a system a generalized T-matrix may be defined which describes the radiation from one of the core regions when another one has been excited. It is then a many channel T-matrix in which the channels are the different disjoint scattering regions. It is shown that the formal properties of this T matrix are the same as for a normal T matrix. In § 2 we review the properties of ordinary scattering theory, and then in § 3 we show that analogous properties for the generalized T matrix hold. An exact expression for the density of particle eigenstates is derived in terms of the positions and scattering matrices of the individual scattering centres. This expression reduces to the standard KKR band structure equation for the infinite regular lattice. We also consider how to construct the density of eigenstates and the charge density for such a system. These latter quantities may be approached in two different ways: the usual way is to consider the scattering material to occupy all space, but from a multiple scattering viewpoint one must consider the total volume of condensed material to be small compared with all space, even if both limit to infinity. It is not obvious that the latter method leads to the same results as the former (formally the density of eigenvalues is identical to the free electron density of eigenvalues in the latter method) and it is shown how the differences in the two approaches are resolved. We also discuss the expansion of some of these results for a perfect lattice. While the usual expansions are pseudo-potential expansions, a manifestly ‘on-energy shell’ expansion is derived which does not contain the arbitrary parameters of the pseudo-potential expansions. Finally, in § 4, we review the most significant contributions of other authors to the theory of multiple scattering.