High-Energy Homogenization of a Multidimensional Nonstationary Schrödinger Equation

IF 1.7 3区物理与天体物理 Q2 PHYSICS, MATHEMATICAL

Russian Journal of Mathematical Physics Pub Date : 2023-12-25 DOI:10.1134/S1061920823040064

M. Dorodnyi

引用次数: 0

Abstract

In \(L_2(\mathbb{R}^d)\), we consider an elliptic differential operator \(\mathcal{A}_\varepsilon \! = \! - \operatorname{div} g(\mathbf{x}/\varepsilon) \nabla + \varepsilon^{-2} V(\mathbf{x}/\varepsilon)\), \( \varepsilon > 0\), with periodic coefficients. For the nonstationary Schrödinger equation with the Hamiltonian \(\mathcal{A}_\varepsilon\), analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator \(\mathcal{A}_1\) are studied (the so called high-energy homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in \(L_2(\mathbb{R}^d)\)-norm for small \(\varepsilon\) are obtained.

DOI 10.1134/S1061920823040064

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多维非稳态薛定谔方程的高能量均质化

Abstract In \(L_2(\mathbb{R}^d)\), we consider an elliptic differential operator \(\mathcal{A}_\varepsilon \！= \!- operatorname{div} g(\mathbf{x}/\varepsilon) \nabla + \varepsilon^{-2} V(\mathbf{x}/\varepsilon)\), \( \varepsilon > 0\), 具有周期性系数。对于具有哈密顿的非稳态薛定谔方程（\(\mathcal{A}_\varepsilon\），研究了与算子\(\mathcal{A}_1\)的离散关系的任意点相关的同质化问题（即所谓的高能同质化）。对于具有特殊初始数据的这些方程的考希问题解，得到了小\(\varepsilon\)时的\(L_2(\mathbb{R}^d)\norm)近似值。 doi 10.1134/s1061920823040064

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来源期刊

Russian Journal of Mathematical Physics 物理-物理：数学物理

CiteScore

3.10

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.