Nonlinear Coherent States and Ehrenfest Time for Schrödinger Equation

We consider the propagation of wave packets for the nonlinear Schrödinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.     

Lyapunov Exponent and Density of States of a One-Dimensional Non-Hermitian Schrödinger Equation

We calculate, using numerical methods, the Lyapunov exponent (E) and the density of states (E) at energy E of a one-dimensional non-Hermitian Schrödinger equation with off-diagonal disorder. For the particular case we consider, both (E) and (E) depend only on the modulus of E. We find a pronounced maximum of (|E|) at energy E=2/ , which seems to be linked to the fixed point structure of an associated random map. We show how the density of states (E) can be expanded in powers of E. We find (|E|)=(1/ ²)+(4/3 ³) |E|²+. This expansion, which seems to be asymptotic, can be carried out to an arbitrarily high order.     

Symplectic Structures for the Cubic Schrödinger Equation in the Periodic and Scattering Case

We develop a unified approach for construction of symplectic forms for 1D integrable equations with the periodic and rapidly decaying initial data. As an example we consider the cubic nonlinear Schrödinger equation. Mathematics Subject Classifications (2000) 35Q53, 58B99.The work is partially supported by NSF grant DMS-9971834.     

Derivation of Nonlinear Schrödinger Equation

We propose some nonlinear Schrödinger equations by adding some higher order terms to the Lagrangian density of Schrödinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schrödinger equation is weak.     

Thermodynamic Gravity and the Schrödinger Equation

We adopt a formulation of the Mach principle that the rest mass of a particle is a measure of it’s long-range collective interactions with all other particles inside the horizon. As a consequence, all particles in the universe form a ‘gravitationally entangled’ statistical ensemble and one can apply the approach of classical statistical mechanics to it. It is shown that both the Schrödinger equation and the Planck constant can be derived within this Machian model of the universe. The appearance of probabilities, complex wave functions, and quantization conditions is related to the discreetness and finiteness of the Machian ensemble.     

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.     

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.     

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.     

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.     

Homogenization of the Schrödinger Equation and Effective Mass Theorems

In this paper we discuss two different models of dependent percolation on the graph ². These models can be thought of as percolation in a random environment. They were inspired by the work of McCoy and Wu [7,8] on the Ising model in a random environment as well as other models of particle systems in a random environment [9, 5, 6, 3]. We show that both models of dependent percolation exhibit phase transitions. This proves a version of stability for percolation on ² and proves a conjecture of Jonasson, Mossel and Peres [4], who proved a similar result on ³.Research supported in part by an NSF postdoctoral fellowshipAcknowledgement I would like to thank David Levin and Yuval Peres for introducing me to the problem. I would also like to thank Yuval Peres and Eric Babson for helpful conversations.     

On Gravitational Effects in the Schrödinger Equation

The Schrödinger  equation for a particle of rest mass $m$ and electrical charge $ne$ interacting with a four-vector potential $A_i$ can be derived as the non-relativistic limit of the Klein–Gordon  equation $\left( \Box '+m^2\right) \varPsi =0$ for the wave function $\varPsi $ , where $\Box '=\eta ^{jk}\partial '_j\partial '_k$ and $\partial '_j=\partial _j -\mathrm {i}n e A_j$ , or equivalently from the one-dimensional  action $S_1=-\int m ds +\int neA_i dx^i$ for the corresponding point particle in the semi-classical approximation $\varPsi \sim \exp {(\mathrm {i}S_1)}$ , both methods yielding the equation $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2m}\eta ^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + m + n e\phi \right) \varPsi $ in Minkowski  space–time  , where $\alpha ,\beta =1,2,3$ and $\phi =-A_0$ . We show that these two methods generally yield equations  that differ in a curved background  space–time   $g_{ij}$ , although they coincide when $g_{0\alpha }=0$ if $m$ is replaced by the effective mass $\mathcal{M}\equiv \sqrt{m^2-\xi R}$ in both the Klein–Gordon  action $S$ and $S_1$ , allowing for non-minimal coupling to the gravitational  field, where $R$ is the Ricci scalar and $\xi $ is a constant. In this case $\mathrm {i}\partial _0\varPsi \approx \left( \frac{1}{2\mathcal{M}'} g^{\alpha \beta }\partial '_{\alpha }\partial '_{\beta } + \mathcal{M}\phi ^{(\mathrm g)} + n e\phi \right) \varPsi $ , where $\phi ^{(\mathrm g)} =\sqrt{g_{00}}$ and $\mathcal{M}'=\mathcal{M}/\phi ^{(\mathrm g)} $ , the correctness of the gravitational  contribution to the potential having been verified to linear order $m\phi ^{(\mathrm g)} $ in the thermal-neutron beam interferometry experiment due to Colella et al. Setting $n=2$ and regarding $\varPsi $ as the quasi-particle wave function, or order parameter, we obtain the generalization of the fundamental macroscopic Ginzburg-Landau equation of superconductivity to curved space–time. Conservation of probability and electrical current requires both electromagnetic gauge and space–time  coordinate conditions to be imposed, which exemplifies the gravito-electromagnetic analogy, particularly in the stationary case, when div ${{\varvec{A}}}=\hbox {div}{{\varvec{A}}}^{(\mathrm g)}=0$ , where ${{\varvec{A}}}^{\alpha }=-A^{\alpha }$ and ${{\varvec{A}}}^{(\mathrm g)\alpha }=-\phi ^{(\mathrm g)}g^{0\alpha }$ . The quantum-cosmological Schrödinger  (Wheeler–DeWitt) equation is also discussed in the $\mathcal{D}$ -dimensional  mini-superspace idealization, with particular regard to the vacuum potential $\mathcal V$ and the characteristics of the ground state, assuming a gravitational  Lagrangian   $L_\mathcal{D}$ which contains higher-derivative  terms up to order $\mathcal{R}^4$ . For the heterotic superstring theory  , $L_\mathcal{D}$ consists of an infinite series in $\alpha '\mathcal{R}$ , where $\alpha '$ is the Regge slope parameter, and in the perturbative approximation $\alpha '|\mathcal{R}| \ll 1$ , $\mathcal V$ is positive semi-definite for $\mathcal{D} \ge 4$ . The maximally symmetric ground state satisfying the field equations is Minkowski  space for $3\le {\mathcal {D}}\le 7$ and anti-de Sitter  space for $8 \le \mathcal {D} \le 10$ .     

A Fractional Schrödinger Equation and Its Solution

This paper presents a fractional Schrödinger equation and its solution. The fractional Schrödinger equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives to obtain the fractional Euler-Lagrange equations of motion. We present the Lagrangian for the fractional Schrödinger equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Schrödinger equation which is the same as that obtained using the fractional variational principle. As an example, we consider the eigensolutions of a particle in an infinite potential well. The solutions are obtained in terms of the sines of the Mittag-Leffler function.     

Decoherence of Photon-Added Schrödinger Cat States in Thermal Environment

We analyze the statistical properties and decoherence of the field states produced by adding any number of photons to the Schrodinger Cat states (SCSs) in a thermal environment. It is found that the normalization factor of PA-SCS is the Hermite polynomial of the coherent factor α. The statistical properties and decoherence is discussed by deriving analytically the time evolution of the Q-function, Wigner function and the photon-number distribution. It is shown that the single photon-added even SCSs, the Wigner function is always positive in the whole phase space when κt exceeds the threshold value $\frac{1}{2}\ln [ ( 2\bar{n}+2 ) / ( 2\bar{n}+1 ) ] $ . This implies that the single photon-added even SCSs possesses a robustness due to the presence of photon-addition, which can also be seen from the point of view of the volume of negative region of Wigner function.     

The Hamiltonian Structure of the Nonlinear Schrödinger Equation and the Asymptotic Stability of its Ground States

In this paper we prove that ground states of the NLS which satisfy the sufficient conditions for orbital stability of M. Weinstein, are also asymptotically stable, for seemingly generic equations. The key issue is to prove that a certain coefficient is non-negative because is a square power. We assume that the NLS has a smooth short range nonlinearity. We assume also the presence of a very short range and smooth linear potential, to avoid translation invariance. The basic idea is to perform a Birkhoff normal form argument on the hamiltonian, as in a paper by Bambusi and Cuccagna on the stability of the 0 solution for NLKG. But in our case, the natural coordinates arising from the linearization are not canonical. So we need also to apply the Darboux Theorem. With some care though, in order not to destroy some nice features of the initial hamiltonian.     

The Vacuum Electromagnetic Fields and the Schrödinger Equation

We consider the simple case of a nonrelativistic charged harmonic oscillator in one dimension, to investigate how to take into account the radiation reaction and vacuum fluctuation forces within the Schrödinger equation. The effects of both zero-point and thermal classical electromagnetic vacuum fields, characteristic of stochastic electrodynamics, are separately considered. Our study confirms that the zero-point electromagnetic fluctuations are dynamically related to the momentum operator p=?i ? ?/? x used in the Schrödinger equation.     

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.