有界区间上薛定谔算子的极大行列式
Maximal determinants of Schrödinger operators on bounded intervals
摘要：
考虑在$L^q$ -范数约束（$q\geq 1$）下，在Dirichlet边界条件下，在有界区间上定义的一维薛定谔算子的泛函行列式的极值势的求解问题.这是通过首先扩展的定义的功能行列式的情况下，$L^q$潜力和显示所产生的问题是等价的问题，在最优控制，我们相信是独立的利益。我们证明存在性，唯一性和描述这个问题的解决方案的一些基本性质的所有$q\geq 1$，提供了一个完整的表征的情况下，$q$是一个（脉冲）和两个（维尔斯特拉斯的$\wp$功能）的极值电位。
英文摘要：
We consider the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrödinger operator defined on a bounded interval with Dirichlet boundary conditions under an $L^q$-norm restriction ($q\geq 1$). This is done by first extending the definition of the functional determinant to the case of $L^q$ potentials and showing the resulting problem to be equivalent to a problem in optimal control, which we believe to be of independent interest. We prove existence, uniqueness and describe some basic properties of solutions to this problem for all $q\geq 1$, providing a complete characterization of extremal potentials in the case where $q$ is one (a pulse) and two (Weierstrass's $\wp$ function).
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