Schrödinger-Type方程的均匀化:带校正器的算子估计

IF 0.6 4区 数学 Q3 MATHEMATICS
T. A. Suslina
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引用次数: 1

摘要

在\(L_2(\mathbb R^d;\mathbb C^n)\)中,我们考虑一个自伴随椭圆二阶微分算子\(A_\varepsilon\)。假设\(A_\varepsilon\)的系数是周期性的,并且依赖于\(\mathbf x/\varepsilon\),其中\(\varepsilon>0\)是一个小参数。我们研究了算子指数\(e^{-iA_\varepsilon\tau}\)对于小的\(\varepsilon\)和\(\tau\in\mathbb R\)的行为。应用所得结果研究了一类具有初始数据的Schrödinger-type方程\(i\partial_\tau \mathbf{u}_\varepsilon(\mathbf x,\tau) = - (A_\varepsilon{\mathbf u}_\varepsilon)(\mathbf x,\tau)\)的Cauchy问题解的性质。对于固定的\(\tau\)和\(\varepsilon\to 0\), \({\mathbf u}_\varepsilon(\,\boldsymbol\cdot\,,\tau)\)的解在\(L_2(\mathbb R^d;\mathbb C^n)\)收敛于均匀化问题的解;错误顺序为\(O(\varepsilon)\)。我们在\(L_2(\mathbb R^d;\mathbb C^n)\)和\(H^1(\mathbb R^d;\mathbb C^n)\)中分别获得了误差为\(O(\varepsilon^2)\)和\(O(\varepsilon)\)的近似解\({\mathbf u}_\varepsilon(\,\boldsymbol\cdot\,,\tau)\)。这些近似包括适当的校正。跟踪了误差对\(\tau\)的依赖关系。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Homogenization of the Schrödinger-Type Equations: Operator Estimates with Correctors

In \(L_2(\mathbb R^d;\mathbb C^n)\) we consider a self-adjoint elliptic second-order differential operator \(A_\varepsilon\). It is assumed that the coefficients of \(A_\varepsilon\) are periodic and depend on \(\mathbf x/\varepsilon\), where \(\varepsilon>0\) is a small parameter. We study the behavior of the operator exponential \(e^{-iA_\varepsilon\tau}\) for small \(\varepsilon\) and \(\tau\in\mathbb R\). The results are applied to study the behavior of the solution of the Cauchy problem for the Schrödinger-type equation \(i\partial_\tau \mathbf{u}_\varepsilon(\mathbf x,\tau) = - (A_\varepsilon{\mathbf u}_\varepsilon)(\mathbf x,\tau)\) with initial data in a special class. For fixed \(\tau\) and \(\varepsilon\to 0\), the solution \({\mathbf u}_\varepsilon(\,\boldsymbol\cdot\,,\tau)\) converges in \(L_2(\mathbb R^d;\mathbb C^n)\) to the solution of the homogenized problem; the error is of order \(O(\varepsilon)\). We obtain approximations for the solution \({\mathbf u}_\varepsilon(\,\boldsymbol\cdot\,,\tau)\) in \(L_2(\mathbb R^d;\mathbb C^n)\) with error \(O(\varepsilon^2)\) and in \(H^1(\mathbb R^d;\mathbb C^n)\) with error \(O(\varepsilon)\). These approximations involve appropriate correctors. The dependence of errors on \(\tau\) is traced.

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来源期刊
CiteScore
0.90
自引率
0.00%
发文量
7
审稿时长
>12 weeks
期刊介绍: Functional Analysis and Its Applications publishes current problems of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. The journal also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.
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