Special Class Of Potentials Research Articles

From two-dimensional (super-integrable) quantum dynamics to (super-integrable) three-body dynamics

It is shown that planar quantum dynamics can be related to three-body quantum dynamics in the space of relative motion with a special class of potentials. As an important special case the O(d) symmetry reduction from d degrees of freedom to one degree is presented. A link between two-dimensional (super-integrable) systems and three-body (super-integrable) systems is revealed. As illustration we present number of examples. We demonstrate that the celebrated Calogero–Wolfes three-body potential has a unique property: two-dimensional quantum dynamics coincides with three-body quantum dynamics on the line at d = 1; it is governed by the Tremblay–Turbiner–Winternitz potential for parameter k = 3.

Random normal matrices and polynomial curves

We show that in the large matrix limit, the eigenvalues of the normal matrix model for matrices with spectrum inside a compact domain with a special class of potentials homogeneously fill the interior of a polynomial curve uniquely defined by the area of its interior domain and its exterior harmonic moments which are all given as parameters of the potential. Then we consider the orthogonal polynomials corresponding to this matrix model and show that, under certain assumptions, the density of the zeros of the highest relevant orthogonal polynomial in the large matrix limit is (up to some constant factor) given by the discontinuity of the Schwarz function of this polynomial curve.

Investigation of the kernel of the evolution operator for one class of potentials

The convergence of the Schwinger–DeWitt expansion in quantum mechanics is investigated for a special class of potentials for which the squared derivative of a potential (e.g., the centrifugal potential, the modified Peschle–Teller potential, or the potential specified by the Weierstrass function) is represented as a polynomial in the powers of the potential itself. It has been established for which representatives of the class the expansion diverges and for which it converges (convergence may take place only at certain discrete values of the charge). For potentials singular at zero, the absence of singularities in the formula for the kernel has been demonstrated. A generalization of potentials with a converging expansion to a multidimensional case is proposed.

POTENTIALS WITH CONVERGENT SCHWINGER-DEWITT EXPANSION

Convergence of the Schwinger–DeWittexpansion for the evolution operator kernel for specialclass of potentials is studied. It is shown that thisexpansion, which is in the general case asymptotic,converges for the potentials considered (widely used, inparticular, in one-dimensional many-body problems), andthat convergence takes place only for definite discretevalues of the coupling constant. For other values of the charge, a divergent expansiondetermines the kernels having essential singularity atthe origin (beyond the usual δ-function). If oneconsiders only this class of potentials, then one can avoid many problems connected withasymptotic expansions, and one gets a theory withdiscrete values of the coupling constant that is incorrespondence with the discreteness of charge innature. This approach can be applied to quantum fieldtheory.

Recovery of singularities from amplitude information

Let V(x) denote a potential in the one-dimensional Schrödinger equation, without bound states, for which (1) V(x)=0 for x<0 and (2) V(x) is piecewise continuous with adequate decay as x goes to infinity. We are interested in the problem of determining V given the reflectivity r(k)=|R(k)|, where R is the usual left-hand reflection coefficient. For very special classes of potentials it is known that r determines V uniquely. Here we show that under much more general (although still restrictive) assumptions, the location and magnitude of discontinuities of V can be determined from r. The nature of the restrictions is related to the behavior of R(k) for k in the upper half of the complex plane.

Chebyshev's recursion: Analytic principles and applications

We study different algebraic and algorithmic constructions related to the scalar product on the space of polynomials defined on the real axis and on the unit circle and to the Chebyshev procedure. A modern version of the Chebyshev recursion ((m)−T-recursion) is applied to check whether the Hankel and Toeplitz quadratic forms are positive definite, to determine the number of real (complex conjugate) roots of the polynomials, to localize the ordering of these roots, and to find bounds for the values of a function on a given set. We also consider the relation between the (m)−T-recursion and the method of moments in the study of Schrodinger operators for special classes of potentials.

Asymptotic behavior for t?? of the solutions of the equation ?xx+u(x,t)?+(?/4)?=0 with a potential u, satisfying the Korteweg-de Vries equation. II

This is the third paper in a series of papers of the authors, devoted to a rigorous investigation of the asymptotic behavior of the solutions of the KdV equation as t→∞. The immediate purpose of the paper is the investigation of the solution ψ of the Schrodinger equation in the neighborhood of the singular point x=3tλ for a special class of potentials, introduced in the previous papers. As it will be proved, in the final analysis this class of potentials describes the asymptotic behavior of the solutions, decreasing for x→∞, of the KdV equation as t→∞. The solution ψ, far from the singular point, has been investigated earlier. In the paper we investigate a series which gives the solution of the Schrodinger equation and we consider the asymptotic properties of this series.

A new method for solving the one-dimensional Schrödinger equation

Abstract As is well known, the solution of the one-dimensional Schrodinger equation enables one to formulate simple models for the description of phenomena in condensed matter physics. This equation can be solved analytically only for particular potentials. In this paper we describe a new method to solve the one-dimensional Schrodinger equation for a special class of potentials. The method presented here is almost analytic and gives a prescription for computer calculations of the eigenfunctions and eigenvalues.

PLANE-WAVE APPROXIMATION FOR THE SCATTERING AMPLITUDE

Different methods of calculating the S-wave scattering amplitude using the plane-wave approximation are discussed. The resulting scattering amplitudes satisfy the unitarity condition and for high energies approach the Born approximation asymptotically. These methods are applied to the special class of potentials that can be expressed as an infinite sum of exponential potentials, and in particular to some potentials for which the exact solutions are known. It is found that imposing the unitarity condition on the Born approximation, in general, does not improve the result of the calculation.