Growth of Sobolev Norms and Controllability of the Schrödinger Equation

In this paper we obtain a stabilization result for both linear and nonlinear Schrödinger equations under generic assumptions on the potential. Then we consider the Schrödinger equations with a potential which has a random time-dependent amplitude. We show that if the distribution of the amplitude is sufficiently non-degenerate, then any trajectory of the system is almost surely non-bounded in Sobolev spaces.     

A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation

We consider the problem of identifying sharp criteria under which radial H ¹ (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) i? _t u + Δu + |u|² u = 0 scatter, i.e., approach the solution to a linear Schrödinger equation as t → ±∞. The criteria is expressed in terms of the scale-invariant quantities ${\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}}We consider the problem of identifying sharp criteria under which radial H ¹ (finite energy) solutions to the focusing 3d cubic nonlinear Schr?dinger equation (NLS) i∂ _t u + Δu + |u|² u = 0 scatter, i.e., approach the solution to a linear Schr?dinger equation as t → ±∞. The criteria is expressed in terms of the scale-invariant quantities and M[u]E[u], where u ₀ denotes the initial data, and M[u] and E[u] denote the (conserved in time) mass and energy of the corresponding solution u(t). The focusing NLS possesses a soliton solution e ^it Q(x), where Q is the ground-state solution to a nonlinear elliptic equation, and we prove that if M[u]E[u] < M[Q]E[Q] and , then the solution u(t) is globally well-posed and scatters. This condition is sharp in the sense that the soliton solution e ^it Q(x), for which equality in these conditions is obtained, is global but does not scatter. We further show that if M[u]E[u] < M[Q]E[Q] and , then the solution blows-up in finite time. The technique employed is parallel to that employed by Kenig-Merle [17] in their study of the energy-critical NLS.     

A Normal Form for the Schrödinger Equation with Analytic Non-linearities

We discuss a class of normal forms of the completely resonant non-linear Schrödinger equation on a torus. We stress the geometric and combinatorial constructions arising from this study.     

Lorentz Transformations for the Schrödinger Equation

Abstract

We show that the free Schrödinger equation admits Lorentz space-time transformations when corresponding transformations of the ψ-function are nonlocal. Some consequences of this symmetry are discussed.

Dedicated to Wilhelm Fushchych – Inspirer, Mentor, Friend and Pioneer in non–Lie symmetry methods – on the occasion of his sixtieth birthday     

Spectral Determinants for Schrödinger Equation and Q-Operators of Conformal Field Theory

Relation between the vacuum eigenvalues of CFT Q-operators and spectral determinants of one-dimensional Schrödinger operator with homogeneous potential, recently conjectured by Dorey and Tateo for special value of Virasoro vacuum parameter p, is proven to hold, with suitable modification of the Schrödinger operator, for all values of p.     

Symmetries of the Free Schrödinger Equation

An algorithm is proposed for studying the symmetry properties of equations used in theoretical and mathematical physics. The application of this algorithm to the free Schrödinger equation permits one to establish that, in addition to the known Galilei symmetry, the free Schrödinger equation possesses also relativistic symmetry in some generalized sense. This property of the free Schrödinger equation provides an extension of the equation into the relativistic domain of the free particle motion under study.     

Schrdinger Equation of a Particle on a Rotating Curved Surface

We derive the Schrdinger equation of a particle constrained to move on a rotating curved surface S.Using the thin-layer quantization scheme to confine the particle on S,and with a proper choice of gauge transformation for the wave function,we obtain the well-known geometric potential V_g and an additive Coriolis-induced geometric potential in the co-rotational curvilinear coordinates.This novel effective potential,which is included in the surface Schrdinger equation and is coupled with the mean curvature of S,contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian.We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.     

Iterative Solutions of the Schrödinger Equation

In the first example containing a long ranged potential, the long range part of the solution is obtained by an iterative Born-series type method. The convergence is illustrated for a case with the long range part of the potential given by C ₆/r ⁶. Accuracies of 1 : 10⁸ are achieved after 8 iterations. The second example iteratively calculates the solution of a non-linear Gross–Pitaevskii equation for condensed Bose atoms contained in a trap at low temperature.     

Exact Solutions of the Schrödinger Equation with Inverse-Power Potential

The Schrödinger equation for stationary states is studied in a central potential V(r) proportional to r ^– in an arbitrary number of spatial dimensions. The presence of a single term in the potential makes it impossible to use previous algorithms, which only work for quasi-exactly-solvable problems. Nevertheless, the analysis of the stationary Schrödinger equation in the neighbourhood of the origin and of the point at infinity is found to provide relevant information about the desired solutions for all values of the radial coordinate. The original eigenvalue equation is mapped into a differential equation with milder singularities, and the role played by the particular case = 4 is elucidated. In general, whenever the parameter is even and larger than 4, a recursive algorithm for the evaluation of eigenfunctions is obtained. Eventually, in the particular case of two spatial dimensions, the exact form of the ground-state wave function is obtained for a potential containing a finite number of inverse powers of r, with the associated energy eigenvalue.     

Exact Solutions of the Schrödinger Equation with Irregular Singularity

The 1D nonrelativistic Schrödinger equation possessing an irregular singularpoint is investigated. We apply a general theorem about existence and structureof solutions of linear ordinary differential equations to the Schrödinger equationand obtain suitable ansatz functions and their asymptotic representations for alarge class of singular potentials. Using these ansatz functions, we work out allpotentials for which the irregular singularity can be removed and replaced by aregular one. We obtain exact solutions for these potentials and present sourcecode for the computer algebra system Mathematica to compute the solutions. Forall cases in which the singularity cannot be weakened, we calculate the mostgeneral potential for which the Schrödinger equation is solved by the ansatzfunctions obtained and develop a method for finding exact solutions.     

Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise

We construct generalized grand-canonical- and canonical Gibbs measures for a Hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear Schrödinger equation with a focusing nonlinearity of order p < 6. Key properties of these Gibbs measures, in particular absence of “phase transitions” and regularity properties of field samples, are established. We then study a time evolution of this system given by the Hamiltonian evolution perturbed by a stochastic noise term that mimics effects of coupling the system to a heat bath at some fixed temperature. The noise is of Ornstein–Uhlenbeck type for the Fourier modes of the field, with the strength of the noise decaying to zero, as the frequency of the mode tends to ∞. We prove exponential approach of the state of the system to a grand-canonical Gibbs measure at a temperature and “chemical potential” determined by the stochastic noise term.     

Instability for the Semiclassical Non-linear Schrödinger Equation

We adapt recent results on instability for non-linear Schrödinger equations to the semi-classical setting. Rather than work with Sobolev spaces we estimate projective instability in terms of the small constant, h, appearing in the equation. Our motivation comes from the Gross-Pitaevski equation used in the study of Bose-Einstein condensation.     

Colliding Solitons for the Nonlinear Schrödinger Equation

We study the collision of two fast solitons for the nonlinear Schrödinger equation in the presence of a slowly varying external potential. For a high initial relative speed ||v|| of the solitons, we show that, up to times of order ||v|| after the collision, the solitons preserve their shape (in L ²-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential.     

From the N-body Schrödinger Equation to the Quantum Boltzmann Equation: a Term-by-Term Convergence Result in the Weak Coupling Regime

In this paper we analyze the asymptotic dynamics of a system of N quantum particles, in a weak coupling regime. Particles are assumed statistically independent at the initial time. Our approach follows the strategy introduced by the authors in a previous work [BCEP1]: we compute the time evolution of the Wigner transform of the one-particle reduced density matrix; it is represented by means of a perturbation series, whose expansion is obtained upon iterating the Duhamel formula; this approach allows us to follow the arguments developed by Lanford [L] for classical interacting particles evolving in a low density regime. We prove, under suitable assumptions on the interaction potential, that the complete perturbation series converges term-by-term, for all times, towards the solution of a Boltzmann equation. The present paper completes the previous work [BCEP1]: it is proved there that a subseries of the complete perturbation expansion converges uniformly, for short times, towards the solution to the nonlinear quantum Boltzmann equation. This previous result holds for (smooth) potentials having possibly non-zero mean value. The present text establishes that the terms neglected at once in [BCEP1], on a purely heuristic basis, indeed go term-by-term to zero along the weak coupling limit, at least for potentials having zero mean. Our analysis combines stationary phase arguments with considerations on the nature of the various Feynman graphs entering the expansion.     

A New Approach to the Exact Solutions of the Effective Mass Schrödinger Equation

Effective mass Schrödinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrödinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. One can also get solution of the effective mass Schrödinger equation starting from the constant mass Schrödinger equation.     

Wave Decoherence for the Random Schrödinger Equation with Long-Range Correlations

In this paper, we study the loss of coherence of a wave propagating according to the Schrödinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze the decoherence phenomena is a properly rescaled Wigner transform of the solution of the random Schrödinger equation. We exhibit anomalous wave decoherence effects at different propagation scales.