Singular Continuous Spectrum for a Class of Substitution Hamiltonians II

We consider discrete one-dimensional Schrödinger operators with aperiodic potentials generated by primitive substitutions. Using the three-block version of Gordon's criterion, we establish purely singular continuous spectrum with probability one provided that the potentials have index greater than three. It is also shown that one cannot use this criterion to prove uniform results.     

Singular Continuous Spectrum for a Class of Substitution Hamiltonians

We consider discrete one-dimensional Schrödinger operators with potentials generated by primitive substitutions. A purely singular continuous spectrum with probability one is established provided that the potentials have a local four-block structure.     

Purely absolutely continuous spectrum for almost Mathieu operators

Using a recent result of Sinai, we prove that the almost Mathieu operators acting onl ²(), (l _Y, )(n) = (l+1)+(l–)+ cos(n+) (n) have a purely absolutely continuous spectrum for almost all a provided that is a good irrational and is sufficiently small. Furthermore, the generalized eigen-functions are quasiperiodic.     

From Classical Hamiltonian Flow to Quantum Theory: Derivation of the Schrödinger Equation

It is shown how the essentials of quantum theory, i.e., the Schrödinger equation and the Heisenberg uncertainty relations, can be derived from classical physics. Next to the empirically grounded quantisation of energy and momentum, the only input is given by the assumption of fluctuations in energy and momentum to be added to the classical motion. Extending into the relativistic regime for spinless particles, this procedure leads also to a derivation of the Klein-Gordon equation. Comparing classical Hamiltonian flow with quantum theory, then, the essential difference is given by a vanishing divergence of the velocity of the probability current in the former, whereas the latter results from a much less stringent requirement, i.e., that only the average over fluctuations and positions on the average divergence be identical to zero.     

Uniform Spectral Properties of One-Dimensional Quasicrystals,II. The Lyapunov Exponent

In this Letter we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schrödinger operators with Sturmian potentials. We apply this method to the transfer matrices in order to study the Lyapunov exponent and the growth rate of eigenfunctions. This gives uniform vanishing of the Lyapunov exponent on the spectrum for all irrational rotation numbers. For irrational rotation numbers with bounded continued fraction expansion, it gives uniform existence of the Lyapunov exponent on the whole complex plane. Moreover, it yields uniform polynomial upper bounds on the growth rate of transfer matrices for irrational rotation numbers with bounded density. In particular, all our results apply to the Fibonacci case.     

A level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials

We propose a level set method for the semiclassical limit of the Schrödinger equation with discontinuous potentials. The discontinuities in the potential corresponds to potential barriers, at which incoming waves can be partially transmitted and reflected. Previously such a problem was handled by Jin and Wen using the Liouville equation – which arises as the semiclassical limit of the Schrödinger equation – with an interface condition to account for partial transmissions and reflections (S. Jin, X. Wen, SIAM J. Num. Anal. 44 (2006) 1801–1828). However, the initial data are Dirac-delta functions which are difficult to approximate numerically with a high accuracy. In this paper, we extend the level set method introduced in (S. Jin, H. Liu, S. Osher, R. Tsai, J. Comp. Phys. 210 (2005) 497–518) for this problem. Instead of directly discretizing the Delta functions, our proposed method decomposes the initial data into finite sums of smooth functions that remain smooth in finite time along the phase flow, and hence can be solved much more easily using conventional high order discretization schemes.     

Wave packet dynamics for a non-linear Schrödinger equation describing continuous position measurements

We investigate time-dependent solutions for a non-linear Schrödinger equation recently proposed by Nassar and Miret-Artés (NM) to describe the continuous measurement of the position of a quantum particle (Nassar, 2013; Nassar and Miret-Artés, 2013). Here we extend these previous studies in two different directions. On the one hand, we incorporate a potential energy term in the NM equation and explore the corresponding wave packet dynamics, while in the previous works the analysis was restricted to the free-particle case. On the other hand, we investigate time-dependent solutions while previous studies focused on a stationary one. We obtain exact wave packet solutions for linear and quadratic potentials, and approximate solutions for the Morse potential. The free-particle case is also revisited from a time-dependent point of view. Our analysis of time-dependent solutions allows us to determine the stability properties of the stationary solution considered in Nassar (2013), Nassar and Miret-Artés (2013). On the basis of these results we reconsider the Bohmian approach to the NM equation, taking into account the fact that the evolution equation for the probability density ρ=|ψ|²$\rho ={\left|\psi \right|}^2$ is not a continuity equation. We show that the effect of the source term appearing in the evolution equation for ρ$\rho$ has to be explicitly taken into account when interpreting the NM equation from a Bohmian point of view.     

The spectrum of a one-dimensional hierarchical model

The spectrum of a discrete Schrödinger operator with a hierarchically distributed potential is studied both by a renormalization group technique and by numerical analysis. A suitable choice of the potential makes it possible to reduce the original problem to a two-dimensional map. Scaling laws for the band-edge energyE _be and for the integrated density of states are predicted together with the global properties of the spectrum. Different scaling regimes are obtained depending on a hierarchy positive parameterR: for R<1/2 the usual scaling laws for the periodic case are obtained, while forR>1/2 the scaling behavior depends explicitly onR.     

Stability Criteria for the Weyl m-Function

This paper presents a new approach to spectral theory for theSchrödinger Operator on the half-line. Solutions of nonlinearRiccati-type equations related to the Schrödinger equation at realspectral parameter are characterised by means of their clusteringproperties as is varied. A family of solutions exhibiting aso-called -clustering property is shown to imply precise estimatesfor the complex boundary value of the Weyl m-function and thespectral measure, and leads to an analysis of the absolutely continuouscomponent of the spectral measure in terms of stability criteria for thecorresponding Riccati equations.     

Perturbation expansion for a one-dimensional Anderson model with off-diagonal disorder

The weak disorder expansion for a random Schrödinger equation with off-diagonal disorder in one dimension is studied. The invariant measure, the density of states, and the Lyapunov exponent are computed. The most interesting feature in this model appears at the band center, where the differentiated density of states diverges, while the Lyapunov exponent vanishes. The invariant measure approaches an atomic measure concentrated on zero and infinity. The results extend previous work of Markos to all orders of perturbation theory.     

受限氢原子光谱高能区的共振增强

通过求解含时薛定谔方程,得到了受限氢原子的光谱结构,发现了明显的共振增强现象,但是在不同的能区限制势对谱线的影响并不一样;此外,还计算了受限氢原子高激发态的抗磁谱受限制势影响而出现的增强现象.     

Continuity of the Lyapunov Exponent for Quasiperiodic Operators with Analytic Potential

We study regularity properties of the Lyapunov exponent L of one-frequency quasiperiodic operators with analytic potential, under no assumptions on the Diophantine class of the frequency. We prove joint continuity of L, in frequency and energy, at every irrational frequency.     

Non-Existence of Liouvillian Solutions for a Free Quantum Particle on a Torus Surface,Part 1: Polar States

We consider a quantum particle constrained to the surface of a torus that we parametrize by its azimuthal and polar angle. We show that the corresponding Schrödinger equation does not have closed-form solutions (in the sense of Liouvillian functions) that depend on the polar angle only. It follows that if there are any wavefunctions in closed form, they must contain nondegenerate, special functions.     

Fresnel Type Path Integral for the Stochastic Schrödinger Equation

A path integral representation is obtained for the stochastic partial differential equation of Schrödinger type arising in the theory of open quantum systems subject to continuous nondemolition measurement and filtering, known as the a posteriori or Belavkin equation. The result is established by means of Fresnel-type integrals over paths in configuration space. This is achieved by modifying the classical action functional in the expression for the amplitude along each path by means of a stochastic Itô integral. This modification can be regarded as an extension of Menski's path integral formula for a quantum system subject to continuous measurement to the case of the stochastic Schrödinger equation.     

Envelope self-similar solutions for the nonautonomous and inhomogeneous nonlinear Schrödinger equation

By means of the similarity transformation connecting with the solvable stationary equation, the self-similar combined Jacobian elliptic function solutions and fractional form solutions of the generalized nonlinear Schrödinger equation (NLSE) are obtained when the dispersion, nonlinearity, and gain or absorption are varied. The propagation dynamics in a periodic distributed amplification system is investigated. Self-similar cnoidal waves and corresponding localized waves including bright and dark similaritons (or solitons) for NLSE and arch and kink similaritons (or solitons) for cubic-quintic NLSE are analyzed. The results show that the intensity and the width of chirped cnoidal waves (or similaritons) change more distinctly than that of chirp-free counterparts (or solitons).     

Adiabatic invariants and asymptotic behavior of Lyapunov exponents of the Schrödinger equation

We give an upper bound for the high-energy behavior of the Lyapunov exponent of the one-dimensional Schrödinger equation. We relate this behavior to the diffrentiability properties of the potential. As an application, this result provides an upper bound for the asymptotic length of the gaps of the Schrödinger equation.     

Energy Levels of Neutral Atoms Via a New Perturbation Method

The energy levels of neutral atoms supported by potential V (r) = -Zexp(-ar)/r (Yukawa potential) are studied, using both dimensional and dimensionless quantities, via a new analytical methodical proposal (devised to solve for nonexactly solvable Schrödinger equation). Using dimensionless quantities, by scaling the radial Hamiltonian through y = Zr and = /Z, we report that the scaled screening parameter is restricted to have values ranging from zero to less than 0.4. On the other hand, working with the scaled Hamiltonian enhances the accuracy and extremely speeds up the convergence of the energy eigenvalues. The energy levels of several new eligible scaled screening parameter values are also reported.     

Nonlinear quantum mechanics as weyl geometry of a classical statistical ensemble

We derive nonlinear relativistic and non-relativistic wave equations for spin-0 and 1/2 particles. For a suitable choice of coupling constants, the equations become linear and Weyl gauge invariant in the spin-0 case. The Dirac particle is much more subtle. When a suitable gauge is chosen and, when the Compton wavelength of the particle is much larger than Planck's length, we recover the standard Dirac equation. Nonlinear corrections to the Schrödinger equation are obtained and these appear as the first-order relativistic corrections to the non-relativistic Hamilton-Jacobi equation. Consequently, we construct nonbilinear homogeneous realizations of anapproximate Galilean symmetry. We put forth the idea that not only a modification of quantum mechanics might be necessary in order to accommodate gravity, but quantum mechanics itself might have a geometrical origin with Planck's constant as the coupling between matter and curvature.1. We thank L. Boya for this remark.2. If we wish to have nodes for stationary states then we must require that has an inflection point at the node, i.e., ² is zero at such node.3. I. Bialynicki-Biruli and J. Mycielski,Ann. Phys. (N. Y.) 100, 62–93 (1976).