Academic Journal
On Convergence to Equilibrium Distribution for Schrödinger Equation: On convergence to equilibrium distribution for Schrödinger equation
| Title: | On Convergence to Equilibrium Distribution for Schrödinger Equation: On convergence to equilibrium distribution for Schrödinger equation |
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| Authors: | Komech, Alexander, Kopylova, Elena, Mauser, Norbert |
| Source: | Markov Processes and Related Fields. 11(1):81-110 |
| Publisher Information: | Polymat, Moscow, 2005. |
| Publication Year: | 2005 |
| Subject Terms: | Cauchy problem, 103019 Mathematical physics, scattering theory, correlation functions, Astronomy, Spaces of measures, convergence of measures, Schrödinger equation, Astronomie, Stochastic methods applied to problems in equilibrium statistical mechanics, Gaussian measure, Schrödinger operator, Schrödinger equation, 1030 Physics, 1030 Physik, random initial solution, Scattering theory for PDEs, 103019 Mathematische Physik, Random linear operators, mixing condition |
| Description: | Summary: Consider the Schrödinger equation with constant or variable coefficients in \(\mathbb R^3\). We study a distribution \(\mu_t\) of a random solution at a time \(t \in \mathbb R\). An initial measure \(\mu_0\) has translation-invariant correlation matrices, zero mean and finite mean density of charge. It also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition. The main result is the convergence of \(\mu_t\) to a Gaussian measure as \(t \to\pm\infty\) which gives the Central Limit Theorem for the Schrödinger equation. The proof for the case of constant coefficients is based on a spectral cutoff and an analysis of long time asymptotics of the solution in the Fourier representation and Bernstein's `room-corridor' argument. The case of variable coefficients is reduced to that of constant ones by a version of a scattering theory for the solutions with infinite charge. |
| Document Type: | Article |
| File Description: | application/xml |
| ISSN: | 1024-2953 |
| Access URL: | https://zbmath.org/2177569 https://ucrisportal.univie.ac.at/de/publications/8e2a60f3-a1a4-42de-ad74-44d36d78668f |
| Accession Number: | edsair.dedup.wf.002..d77a1d52f151a19f1b5e184f07af05bf |
| Database: | OpenAIRE |
| ISSN: | 10242953 |
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