Two-step minimization approach to an $L^\infty$-constrained variational problem with a generalized potential

TL;DR

This work addresses an $L^\infty$-constrained variational problem on $H^1({\mathbb R})$ for a generalized potential $V\in X$ that includes unbounded, non-positive, and measure-valued terms. It extends the two-step minimization approach from positive bounded potentials to this broad class by developing a decomposition principle and accompanying comparison and perturbation theorems, enabling precise analysis of minimizers and the Sobolev-type constant $m(V)^{-1/2}$. The authors establish existence and qualitative properties of minimizers, continuity of the first-step minimum, and concrete results for Dirac-delta-type potentials and trapped modes in potential wells. Overall, the paper provides rigorous tools to obtain exact Sobolev-type constants under broad potentials and to predict trapped modes, with potential implications for waveguides and quantum systems.

Abstract

We study a variational problem on $H^1({\mathbb R})$ under an $L^\infty$-constraint related to Sobolev-type inequalities for a class of generalized potentials, including $L^p$-potentials, non-positive potentials, and signed Radon measures. We establish various essential tools for this variational problem, including the decomposition principle, the comparison principle, and the perturbation theorem, which are the basis of the two-step minimization method. As for their applications, we present precise results for minimizers of minimization problems, such as the study of potentials of Dirac's delta measure type and the analysis of trapped modes in potential wells.

Theorems & Definitions (15)

• Theorem 2.1: decomposition principle
• Theorem 3.1: AKOp1
• Theorem 3.2: AKOp1
• Theorem 4.1: comparison principle of $m(V)$
• Corollary 4.2
• Theorem 4.3
• Theorem 4.4
• Theorem 4.5: Kametaka et al. Kam08
• Lemma 5.1
• Theorem 5.2