Relationship Between Two Classes of Shape-Invariant Potentials

We show that two classes of shape-invariant potentials are interrelated to each other. For all one-dimensional shape-invariant potentials with parameters related by translation, i.e. the first class of shapc-invariant potentials (SIP1),we can find their multi-parameter deformations with q acting as the deformation parameter, i.e. the second class of shape-invariant potentials (SIP2) with parameters related by scaling. In order to get closed solution of SIP2, we consider two infinitesimal intervals, one is close to q= 0 another close to q = 1, and show that in these intervals we can get separately two first-order approximate solutions in closed form, furthermore we prove that all SIP1 can be obtained by the limiting procedures for corresponding SIP2.``     

Mapping of Shape Invariant Potentials by the Point Canonical Transformation

In this paper by using the method of point canonical transformation we find that the Coulomb and Kratzer potentials can be mapped to the Morse potential. Then we show that the Pöschl-Teller potential type I belongs to the same subclass of shape invariant potentials as Hulthén potential. Also we show that the shape-invariant algebra for Coulomb, Kratzer, and Morse potentials is SU(1,1), while the shape-invariant algebra for Pöschl-Teller type I and Hulthén is SU(2).     

Generation of a complete set of additive shape-invariant potentials from an Euler equation

In supersymmetric quantum mechanics, shape invariance is a sufficient condition for solvability. We show that all conventional additive shape-invariant superpotentials that are independent of ? can be generated from two partial differential equations. One of these is equivalent to the one-dimensional Euler equation expressing momentum conservation for inviscid fluid flow, and it is closed by the other. We solve these equations, generate the set of all conventional shape-invariant superpotentials, and show that there are no others in this category. We then develop an algorithm for generating all additive shape-invariant superpotentials including those that depend on ? explicitly.     

A New Type of Shape-Invariant Potential

It has been shown that Hulthen potental and isotonic harmonic oscillator potential are shape-invariant potentials with a translation of parameters, and isotonic harmonic oscillator potential and three-dimensional harmonic oscillator potential belong to a type of shape-invariant potentials.     

Shape-invariant hypergeometric type operators with application to quantum mechanics

A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. The associated special functions are eigenfunctions of some shape-invariant operators. These operators can be analysed together and the mathematical formalism we use can be extended in order to define other shape-invariant operators. All the shape-invariant operators considered are directly related to Schrödinger-type equations.     

Minimizing the Ground State Energy of an Electron in a Randomly Deformed Lattice

We provide a characterization of the spectral minimum for a random Schrödinger operator of the form \({H = -\Delta + \sum_{i \in \mathbb{Z}^d}q(x - i - \omega_i)}\) in \({L^2(\mathbb{R}^d)}\) , where the single site potential q is reflection symmetric, compactly supported in the unit cube centered at 0, and the displacement parameters ω _i are restricted so that adjacent single site potentials do not overlap. In particular, we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible 2 ^d -clusters of single site potentials.The main tool to prove this is a quite general phenomenon in the spectral theory of Neumann problems, which we dub “bubbles tend to the boundary.” How should a given compactly supported potential be placed into a bounded domain so as to minimize or maximize the first Neumann eigenvalue of the Schrödinger operator on this domain? For square or rectangular domains and reflection symmetric potentials, we show that the first Neumann eigenvalue is minimized when the potential sits in one of the corners of the domain and is maximized when it sits in the center of the domain. With different methods we also show a corresponding result for smooth strictly convex domains.     

Identity for the Exponential-Type Molecule Potentials and the Supersymmetry Shape Invariance

The identity and the supersymmetry shape invariance for a class of exponential-type molecule potentials are studied by introducing a deformed five-parameter exponential-type potential (DFPEP) and via the multi-parameter deformations. It has been shown that the DFPEP is a shape-invariant potential with a translation of parameters. By making use of the shape invariance approach, the exact energy levels are determined for the bound states with zero angular momentum. A class of molecule potentials and their exact energy spectra for the zero angular momentum states are reduced from the DFPEP and a general energy spectrum formula, respectively. The interrelations for some molecule potentials are also discussed.     

Supersymmetry, PT-symmetry and spectral bifurcation

We demonstrate that large class of PT-symmetric complex potentials, which can have isospectral real partner potentials, possess two different superpotentials. In the parameter domain, where the superpotential is unique, the spectrum is real and shape-invariant, leading to translational shift in a suitable parameter by real units. The case of two different superpotentials, leading to same potential, yields broken PT-symmetry, the energy spectra in the two phases being separated by a bifurcation. Interestingly, these two superpotentials generate the two disjoint sectors of the Hilbert space. In the broken case, shape invariance produces complex parametric shifts.     

Invariant Measures for Cherry Flows

We investigate the invariant probability measures for Cherry flows, i.e. flows on the two-torus which have a saddle, a source, and no other fixed points, closed orbits or homoclinic orbits. In the case when the saddle is dissipative or conservative we show that the only invariant probability measures are the Dirac measures at the two fixed points, and the Dirac measure at the saddle is the physical measure. In the other case we prove that there exists also an invariant probability measure supported on the quasi-minimal set, we discuss some situations when this other invariant measure is the physical measure, and conjecture that this is always the case. The main techniques used are the study of the integrability of the return time with respect to the invariant measure of the return map to a closed transversal to the flow, and the study of the close returns near the saddle.     

Effects of complex parameters on classical trajectories of Hamiltonian systems

Anderson et al have shown that for complex energies, the classical trajectories of real quartic potentials are closed and periodic only on a discrete set of eigencurves. Moreover, recently it was revealed that when time is complex t \((t=t_{r}\mathrm {e}^{i\theta _{\tau }}),\) certain real Hermitian systems possess close periodic trajectories only for a discrete set of values of ?? _τ . On the other hand, it is generally true that even for real energies, classical trajectories of non-PT symmetric Hamiltonians with complex parameters are mostly non-periodic and open. In this paper, we show that for given real energy, the classical trajectories of complex quartic Hamiltonians H=p ²+a x ⁴+b x ^k (where a is real, b is complex and k=1 or 2) are closed and periodic only for a discrete set of parameter curves in the complex b-plane. It was further found that the given complex parameter b, the classical trajectories are periodic for a discrete set of real energies (i.e., classical energy gets discretized or quantized by imposing the condition that trajectories are periodic and closed). Moreover, we show that for real and positive energies (continuous), the classical trajectories of complex Hamiltonian H = p ² + μx ⁴, μ = μ _r e ^i?? ) are periodic when ??=4 tan^?1[(n/(2m+n))] for ? n and \(m\in \mathbb {Z}\) .     

Quantum aspects of the equivalence principle

Two thought experiments are discussed which suggest, first, a geometric interpretation of the concept of a (say, vector) potential (i.e., as a kinematic quantity associated with a transformation between moving frames of reference suitably related to the problem) and, second, that, in a quantum treatment one should extend the notion of the equivalence principle to include not only the equivalence of inertial forces with suitable real forces, but also the equivalence of potentials of such inertial forces and the potentials of suitable real forces. The two types of cancellation are physically independent of each other, because of the Aharonov-Bohm effect. Finally, we show that the latter effect itself can be understood geometrically as a kinematic effect arising upon the transformation between the two reference frames.On leave of absence from the Department of Physics, Tel-Aviv University, Israel, and the Department of Physics, Yeshiva University, New York.Supported by the NSF under Contract GP-14911.     

What’s the point? Hole-ography in Poincaré AdS

In the context of the AdS/CFT correspondence, we study bulk reconstruction of the Poincaré wedge of AdS\(_3\) via hole-ography, i.e., in terms of differential entropy of the dual CFT\(_2\). Previous work had considered the reconstruction of closed or open spacelike curves in global AdS, and of infinitely extended spacelike curves in Poincaré AdS that are subject to a periodicity condition at infinity. Working first at constant time, we find that a closed curve in Poincaré is described in the CFT by a family of intervals that covers the spatial axis at least twice. We also show how to reconstruct open curves, points and distances, and obtain a CFT action whose extremization leads to bulk points. We then generalize all of these results to the case of curves that vary in time, and discover that generic curves have segments that cannot be reconstructed using the standard hole-ographic construction. This happens because, for the nonreconstructible segments, the tangent geodesics fail to be fully contained within the Poincaré wedge. We show that a previously discovered variant of the hole-ographic method allows us to overcome this challenge, by reorienting the geodesics touching the bulk curve to ensure that they all remain within the wedge. Our conclusion is that all spacelike curves in Poincaré AdS can be completely reconstructed with CFT data, and each curve has in fact an infinite number of representations within the CFT.     

On the bonding of small group 12 clusters

Characteristic properties as well as possible differences in bonding of small group 12 clusters M_n{\rm M}_n (M = Zn{\rm M} = {\rm Zn}, Cd, Hg; n = 2, ?, 6n = 2, \ldots, 6) have been investigated by quantum chemical ab initio methods, i.e., relativistic large-core pseudopotentials, core-polarization potentials and coupled-cluster correlation treatments. A comparison of cohesive energies and spectroscopic properties like ionization potentials, electron affinities, and vibrational frequencies reveals a close similarity between the clusters of Cd and Hg. For Zn clusters we observed an exceptional increase in stability between Zn₃\rm Zn_3 and Zn₄\rm Zn_4. In order to get a more qualitative picture of the covalent contributions to bonding we have calculated the electron localization function (ELF). The ELF analysis is in accordance with the calculated spectroscopic properties and shows predominant van der Waals interactions with weak covalent contributions for all the cluster sizes considered.     

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.     

双参数形变谐振子奇偶相干态的高阶压缩特性

研究了双参数qs形变谐振子奇偶相干态中光场的高阶压缩特性,并用数值计算方法定量研究了形变参数q和s对这一特性的影响规律.结果表明,qs形变谐振子奇偶相干态均可能呈现奇次方阶压缩效应却无偶次方阶压缩效应,这与谐振子没有形变时的通常情况是不同的.当q和s取一定值时,在qs形变相干态中谐振子强度r²取值的一定范围内,这种反常特性的范围随着q偏离1越大s取值越小而变大.有关单参数q形变奇偶相干态的结论作为特例包含在本文的一般结论之中.     

Detection of new acoustic signals from the human head

New acoustic signals generated in the human head have been found using piezoelectric transducers of longitudinal acoustic oscillations; these signals clearly manifest themselves in recording from the temporal regions and are less pronounced when recording from the forehead. They have a form of 4-ms pulses with a repetition frequency varying from 60 to 120 ms. The signal level exceeds that of thermal acoustic fluctuations by about two orders of magnitude. The signals are formed when the subject of an experiment is relaxed; i.e., the human is at rest with closed eyes and is about to fall asleep. The repetition frequency of these signals is close to the alpha-rhythm frequencies; however, there is no exact correlation between these types of signals at long (of about minute) time intervals. The signals recorded from different temples generally have no strong synchronization. Since some parameters of these signals are close to those of electroencephalographic signals, the former can be referred to as acoustoencephalographic signals.     

Phase Transitions for Uniformly Expanding Maps

Given a uniformly expanding map of two intervals we describe a large class of potentials admitting unique equilibrium measures. This class includes all Hölder continuous potentials but goes far beyond them. We also construct a family of continuous but not Hölder continuous potentials for which we observe phase transitions. This provides a version of the example in (9) for uniformly expanding maps.     

Critical behavior of the Chayes-Machta-Swendsen-Wang dynamics

We study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z >or= alpha/nu is close to but probably not sharp in d = 2 and is far from sharp in d = 3, for all q. The conjecture z >or= beta/nu is false (for some values of q) in both d = 2 and d = 3.     

反应渗流式表面生长的动力学和标度行为

研究了关于A，C两种粒子反应渗流式直线类表面生长的动力学和标度行为．当C粒子的概率q=0时，系统表现出一般的单粒子直线生长．当q≠0时，表面表现出粗化的形貌特性，并发现在高维时存在着粗化转变．当q接近渗流阈值时，系统具有很好的标度特性． 关键词：     

BeAl2O4:Cr3+紫外光谱的研究

在强场理论的基础上,本文考虑了组态相互作用,拟合了各能级位置,并对其紫外光谱进行了讨论.