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Dalian Ligong Daxue xuebao

Stochastic Schrdinger equation driven by pure jump Lévy white noise

Enmin Feng

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The white noise approach for stochastic Schrdinger equation(SSE) driven by pure jump Levy white noise is presented.The potential of the SSE is proportional to the Wick exponential of pure jump Levy white noise,which describes the physical system with jump.The white noise approach is based on (S)-1-characterization theorem.The SSE is firstly reduced to the ordinary non-random Schrdinger equation(OSE) by Hermite transform,which can be solved by Feynmann-Kac formula.Then the solution of the SSE is obtained by(S)-1-characterization theorem,converting the solution of the OSE to a(S)-1-process.Furthermore,the solution is in L1(υ) in sense of weak distribution under some certain conditions.

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The white noise approach for stochastic Schrdinger equation(SSE) driven by pure jump Levy white noise is presented.The potential of the SSE is proportional to the Wick exponential of pure jump Levy white noise,which describes the physical system with jump.The white noise approach is based on (S)-1-characterization theorem.The SSE is firstly reduced to the ordinary non-random Schrdinger equation(OSE) by Hermite transform,which can be solved by Feynmann-Kac formula.Then the solution of the SSE is obtained by(S)-1-characterization theorem,converting the solution of the OSE to a(S)-1-process.Furthermore,the solution is in L1(υ) in sense of weak distribution under some certain conditions.
White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.
The master equation of the quantum noise theory is derived by means of Feynman's path integral method. We propose an equivalent s ochastic process where the wavefunction satisfies a nonlinear SchrOdinger equation except for random moments at which it shows orthogonal jumps. Since the early works [ 1-3] on the theory of quantum noise, great interest has been shown in quantum-me-chanical systems affected by random external forces and also new aspects have appeared [4-6]. Usually, the quan-tum state of such systems i described by a density operator obeying a deterministic differential equation (master equation). If we attribute a state vector (instead of the density operator) to the given system, then the state vector should be considered as a stochastic variable governed by a certain stochastic process. In the first part of our work we derive the master equation for the density operator by a seemingly new applica-tion of the path integral technique. Then, we construct a stochastic process for the wavefunction of the system, while retaining its equivalence with the density operator formalism. In the final part we shortly discuss favoured features of the above stochastic process proposed for the wavefunction. This work is an anticipation of a more general and lengthy explanation of the stochastic properties of the wave-function in the presence of random forces. Now, we are going to introduce the notion of stochastic potential of white-noise type. Here we shall investigate the effect of gaussian white-noise potentials. For brevity, we restrict ourselves to the case of a single point.like par-ticle moving in one dimension. Let us assume that the potential V(x, t) acting on the particle is a stochastic variable (x, t stand for the coordi-nate and time). We define the probability distribution of V by the following generator functional G [h]:
We develop a Schrödinger-picture formulation for a scalar quantum field driven by a Lorentz-invariant white-noise field. The quantum state of the system is described by a stochastic wave functional that evolves according to a stochastic Schrödinger equation. We show that the Gaussian structure of the wave functional is preserved under the stochastic evolution, allowing the dynamics to be reduced to a set of equations for the corresponding kernel functions. These kernel equations are derived and solved exactly, yielding an explicit time-dependent expression for the wave functional. The exact solution enables a direct analysis of the statistical properties of the quantum field in the space of field configurations. In particular, we show that the expectation value of the field operator obeys the same stochastic equation as the classical field obtained from the Euler-Lagrange equation of the action. We further compute the energy density from the stochastic wave functional and evaluate its ensemble average over noise realizations. The resulting energy production rate coincides with that obtained from the corresponding Lindblad equation. This result indicates that the stochastic quantum state remains well defined even though certain derived observables exhibit ultraviolet divergences associated with the white-noise idealization.