Accuracy of Semiclassical Methods for Shape Invariant Potentials

Construction of 2nd stage shape invariant potentials

We introduce concept of next generation shape invariance and show that the process of shape invariant extension can be continued indefinitely.

Semiclassical analysis and sensitivity to initial conditions

We present several recent results concerning the transition between quantum and classical mechanics, in the situation where the underlying dynamical system has an hyperbolic behaviour. The special role of invariant manifolds will be emphasized, and the long time evolution will show how the quantum non-determinism and the classical chaotic sensitivity to initial conditions can be compared, and in a certain sense overlap.

Exactness of SWKB for Shape Invariant Potentials

The supersymmetry based semiclassical method (SWKB) is known to produce exact spectra for conventional shape invariant potentials. In this paper we prove that this exactness follows from their additive shape invariance.

SWKB and proper quantization conditions for translationally shape invariant potentials

Using a recently proposed classification for the primary translationally shape invariant potentials, we show that the exact quantization rule formulated by Ma and Xu is equivalent to the supersymmetric JWKB quantization condition. The energy levels for the two considered categories of shaped invariant potentials are also derived.

Shape Invariant Potentials for Effective Mass Schrodinger Equation

Using the method of shape invariant potentials, a number of exact solutions of one dimensional effective mass Schrodinger equation are obtained. The solutions with equi-spaced spectrum are discussed in detail.

Minimal length in an orbit closure as a semiclassical limit

Consider the action of a connected complex reductive group on a finite-dimensional vector space. A fundamental result in invariant theory states that the orbit closure of a vector v is separated from the origin if and only if some homogeneous invariant polynomial is nonzero on v, i.e. v is not in the null cone. Thus, efficiently finding the minimum distance between the orbit closure and the origin can lead to deterministic algorithms for null cone membership, an important polynomial identity testing problem including the non-commutative Edmonds problem. This connection to optimization has rece…

量子混动的周期轨道轨迹公式：正常双曲不变流形的作用

A Periodic Orbit Trace Formula for Quantum Scrambling: The Role of the Normally Hyperbolic Invariant Manifold

本文推导了局部微canonical OTOC的 semiclassical 扩展，揭示了正常双曲不变流形上不稳定周期轨道对量子混动率的影响，并提出了局部不稳定性指数Λ(J)。

德西特空间可观测量代数上的半经典态与时间反演对称性

Semiclassical state on algebra of observables for de Sitter space and time reversal symmetry

本文针对德西特静态片全息中迹态无法重现半经典体关联函数的问题，引入重时钟态构造，结合时间反演对称性，推导观测者哈密顿量，为观测者模型提供场论基础。

Shape Invariant Natanzon Potentials from Potential Algebra

Using the underlying algebraic structures of Natanzon potentials, we discuss conditions that generate shape invariant potentials. In fact, these conditions give all the known shape invariant potentials corresponding to a translational change of parameters. We also find that while the algebra for the general Natanzon potential is $SO(2,2)$, a subgroup $SO(2,1)$ suffices for all the shape invariant problems of Natanzon type.

'Divide-and-conquer' semiclassical molecular dynamics: An application to water clusters

We present an investigation of vibrational features in water clusters performed by means of our recently established divide-and-conquer semiclassical approach [M. Ceotto, G. Di Liberto, and R. Conte, Phys. Rev. Lett. 119, 010401 (2017)]. This technique allows us to simulate quantum vibrational spectra of high-dimensional systems starting from full-dimensional classical trajectories and projection of the semiclassical propagator onto a set of lower dimensional subspaces. The potential energy surface employed is a many-body representation up to three-body terms, in which monomers and two-body in…

Ma-Xu quantization rule and exact WKB condition for translationally shape invariant potentials

For translationally shape invariant potentials, the exact quantization rule proposed by Ma and Xu is a direct consequence of exactness of the modified WKB quantization condition proved by Barclay. We propose here a very direct alternative way to calculate the appropriate correction for the whole class of translationally shape invariant potentials.

Bäcklund--Darboux transformations of traslationally shape-invariant potentials

Recently, Quesne on the one side and on the other side Bougie, Gangopadhyaya and Mallow, have discovered new translational shape invariant potentials not present in previous classifications. By using ordinary Bäcklund--Darboux transformations we propose a new framework in which such shape invariant potentials can be accommodated.

Solvable rational extension of translationally shape invariant potentials

Combining recent results on rational solutions of the Riccati-Schrödinger equations for shape invariant potentials to the scheme developed by Fellows and Smith in the case of the one dimensional harmonic oscillator, we show that it is possible to generate an infinite set of solvable rational extensions for every translationally shape invariant potential of the second category.

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 considered shape invariant operators are directly related to Schrodinger type equations.