优化量子图的基本特征值间隙

IF 2 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL
Mohammed Ahrami, Zakaria El Allali, Evans M Harrell II, James B Kennedy
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引用次数: 0

摘要

我们研究了用凸势能或适当指定子集上的 "单井 "势能最小化或最大化度量图上薛定谔算子的基本谱隙的问题。(在凸的情况下,我们发现最小化和最大化势是片断线性的,只有有限个非光滑点,但给出的例子表明最优势不一定是常数。这与通常在区间和域上的情况大相径庭,在区间和域上,常数势通常是最小的。在单井情况下,我们证明了最优势是片断常数,具有有限次的跳跃,并在这两种情况下分别给出了最小化势可能具有的非平稳点(跳跃)数量的明确估计。此外,我们还证明,与域不同的是,在给定的类别中,一般不可能仅从图的直径方面找到基本差距的非微观约束。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Optimizing the fundamental eigenvalue gap of quantum graphs
We study the problem of minimizing or maximizing the fundamental spectral gap of Schrödinger operators on metric graphs with either a convex potential or a ‘single-well’ potential on an appropriate specified subset. (In the case of metric trees, such a subset can be the entire graph.) In the convex case we find that the minimizing and maximizing potentials are piecewise linear with only a finite number of points of non-smoothness, but give examples showing that the optimal potentials need not be constant. This is a significant departure from the usual scenarios on intervals and domains where the constant potential is typically minimizing. In the single-well case we show that the optimal potentials are piecewise constant with a finite number of jumps, and in both cases give an explicit estimate on the number of points of non-smoothness, respectively jumps, the minimizing potential can have. Furthermore, we show that, unlike on domains, it is not generally possible to find nontrivial bounds on the fundamental gap in terms of the diameter of the graph alone, within the given classes.
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来源期刊
CiteScore
4.10
自引率
14.30%
发文量
542
审稿时长
1.9 months
期刊介绍: Publishing 50 issues a year, Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.
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