Inequalities on the number of bound states in oscillating potentials

For short-range oscillating potentialsV(r), such that possesses some regularity properties we establish inequalities on the number of bound states. In particular we show that by replacingV(r) by –4(W(r))² in the classical inequalities we get bounds for this new class of potentials. Optimal bounds are also obtained. The behaviour for large coupling constants is studied.     

The inverse scattering problem for singular oscillating potentials

In this paper we show how to apply the Gel'fand-Levitan equations with singular oscillating potentials. We give the general procedure for solving these equations and we illustrate the method by some examples.     

Probing quasiparticle states bound by disparate periodic potentials

Thin films of Ag(111) with two-dimensional crystallinity of large lateral coherence grow on Ge(111), free of in-plane registry with the underlying substrate. Ag s-p electrons forming two-dimensional quantum well states scatter coherently at the buried interface potential, resulting in an unexpected set of new quasiparticle states, as observed by angle-resolved photoemission. These new features originate from interactions among Ag quantum well bands, gaining a momentum equivalent to a reciprocal vector of the substrate lattice.     

A Coulomb-related family of potentials having zero-energy bound states

We propose here a new large class of singular central potentials having zero-energy bound states for many values of angular momenta. The potentials are shown to be closely related to the standard attractive Coulomb interaction. Some of them admit the E≠0$E\ne 0$ bound states as well. A quantum–classical correspondence is also discussed.     

Heavy-quark bound states in potentials with the Bethe-Salpeter equation

The spinless Bethe—Salpeter equation is solved for three attractive static quark-antiquark potentials of the form V(r)=–ar^–+br+c, 01, and the effective non-Coulombic power-law potential of the formV(r)=ar ^0.1+c to obtain the spin-averaged energy levels in bottomonium (b ) and charmonium (c ) families. The shifted 1/N expansion technique is used. Calculations of the energy eigenvalues are carried out up to third order and parameters of each potential are adjusted to obtain the best agreement with the experimental spin-averaged data (SAD). Flavor-dependent and flavor-independent cases are considered in this work.     

Threshold conditions and bound states for locally periodic delta potentials

We present a systematic study of the conditions for the generation of threshold energy eigen states and also the energy spectrum generated by two types of locally periodic delta potentials each having the same strength λV and separation distance parameter a: (a) sum of N attractive potentials and (b) sum of pairs of attractive and repulsive potentials. Using the dimensionless parameter g = λV a in case (a) the values of g = g _n , n = 1, 2, …, N at which threshold energy bound state gets generated are shown to be the roots of Nth order polynomial D ₁(N, g) in g. We present an algebraic recursive procedure to evaluate the polynomial D ₁(N, g) for any given N. This method obviates the need for the tedious mathematical analysis described in our earlier work to generate D ₁(N, g). A similar study is presented for case (b). Using the properties of D ₁(N, g) we establish that in case (a) the critical minimum value of g which guarantees the generation of the maximum possible number of bound states is g = 4. The corresponding result for case (b) is g = 2. A typical set of numerical results showing the pattern of variation of g _n as a function of n and several interesting features of the energy spectrum for different values of g and N are also described.     

Computation of bound states for 2D potentials using discrete basis

Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate in closed form the matrix representation of the associated Hamiltonian for two exactly solvable 2D potentials. This enabled us to treat analytically the full Hamiltonian and compute the associated bound states spectrum as the eigenvalues of the associated analytical matrix representing their Hamiltonians. Finally we compared our results satisfactorily with those obtained using the Gauss quadrature numerical integration approach.

PACS numbers: 03.65.Ge, 34.20.Cf, 03.65.Nk, 34.20.Gj     

Hulthén势任意l波束缚态的近似解析解

在任意l波的离心项1/r2用δ2e-δr/(1-e-δr)2近似表达的条件下,对Hulthen势的径向Schrodinger方程作自变量指数变换,使此转化为超几何微分方程,获得了Hulthen势任意l波束缚态的解析解.给出了解析的能谱方程和用超几何多项式表示的归一化的径向波函数,讨论了近似解析解的意义.     

G-factors of hole bound states in spherically symmetric potentials in cubic semiconductors

Holes in cubic semiconductors have effective spin 3/2 and very strong spin orbitinteraction. Due to these factors properties of hole bound states are highly unusual. Weconsider a single hole bound by a spherically symmetric potential, this can be an acceptoror a spherically symmetric quantum dot. Linear response to an external magnetic field ischaracterized by the bound state Lande g-factor. We calculate analytically g-factors of all boundstates.     

Some properties of bound states in the three-body problem with separable potentials

A system of three identical bosons with the pair separable S-interaction is considered. The monotonic (Yamaguchi potential) and oscillating pair potentials for the three-alpha model of¹²C have been investigated. In the case of the Yamaguchi potential many excited states (second, third, etc.) have been found.The author is thankful to Prof. V. G. Neudatchin and Prof. Yu. F. Smirnov for valuable discussions on the problems concerned.     

Wave operators for oscillating potentials

We consider the Schrödinger operator with magnetic vector potential and static scalar potential. We show the existence of the wave operators under considerations which allow strong oscillations of the potentials.     

The Dirac operator with highly singular potentials

Time-dependent scattering theory for a Dirac particle with highly singular potential is developed. Criteria for asymptotic completeness of wave operators are obtained, and an example is given of a potential which violates asymptotic completeness and the unitarity of the scattering operator. (Completeness breaks down for a regular sequence of values of the coupling constant.)     

Estimates of the number of quantal bound states in one and three dimensions

Using the relation between the number of bound states and the number of zeros of the radial eigen-functionψ(r), or equivalently, that ofφ(r)=rψ(r) in the range 0⩽r⩽∞, the upper bounds on the number of bound states generated by potentialV(r) in different angular momentum channels are obtained in three dimension. Using a similar procedure, the upper bound on the number of bound states in one dimension is also deduced. The analysis is restricted to a class of potentials for whichE=0 is the threshold. By taking a number of specific examples, it is demonstrated that both in one and three dimensions, the estimate of the upper bound obtained by this procedure is very close to or equal to the exact number of bound states. The correlation of the present method with the Levison’s theorem and WKB approximation is discussed.     

New estimates on the number of bound states of Schrödinger operators

In this Letter we prove inequalities on the number of bound states of Schrödinger operators by improving the method of Li and Yau relying on heat kernel estimates by Sobolev inequalities. We extend the range of applications and obtain better estimates. In particular, our technique yields a simple proof of Bargmann's bound and very good numerical values.     

The generalized pseudospectral approach to the bound states of the Hulthén and the Yukawa potentials

The generalized pseudospectral (GPS) method is employed to calculate the bound states of the Hulthén and the Yukawa potentials in quantum mechanics, with special emphasis onhigher excited states andstronger couplings. Accurate energy eigenvalues, expectation values and radial probability densities are obtained through a non-uniform and optimal spatial discretization of the radial Schrödinger equation. Results accurate up to thirteen to fourteen significant figures are reported for all the 55 eigenstates of both these potentials withn <- 10 for arbitrary values of the screening parameters covering a wide range of interaction. Furthermore, excited states as high asn = 17 have been computed with good accuracy for both these potentials. Excellent agreement with the available literature data has been observed in all cases. Then > 6 states of the Yukawa potential has been considerably improved over all other existing results currently available, while the same for Hulthén potential are reported here for the first time. Excepting the 1s and 2s states of the Yukawa potential, the present method surpasses the accuracy of all other existing results in the stronger coupling region for all other states of both these systems. This offers a simple and efficient scheme for the accurate calculation of these and other screened Coulomb potentials.     

Franck-Condon factor phase method for determining the potentials of bound states of piatomic molecules

A new method for determining a fragment of the potential function of a bound state of a piatomic molecule from the phase of the matrix element of an electronic transition is proposed. As opposed to the method of WKB nodes [Opt. Spektrosk. 83, 906 (1997)], the new method (we call it the Franck-Condon factor phase method) is suitable for analyzing only those spectral regions free from interference structures. However, it is less demanding to the quality of experimental data and makes it possible to study situations where only a portion of the electronic band profile is accessible for analysis. Accuracy characteristics of the method are demonstrated using the analysis of the spectra of the transitions in Na₂(2³Σ _g ⁺ (v=15, N=27; v=5, N=39)→³Σ _u ⁺ ) as an example.     

Supersymmetry and singular potentials

The breaking of supersymmetry due to singular potentials in supersymmetric quantum mechanics is critically analyzed. It is shown that, when properly regularized, these potentials respect supersymmetry, even when the regularization parameter is removed.     

The properties of scalar quark bound states

We discuss the (static) potential for scalar quark bound states, their leptonic and hadronic decays and their radiative transitions.