On some a priori majorant of eigenvalues of Sturm--Liouville problems

TL;DR

This paper establishes a new upper bound for the first eigenvalues of certain Sturm--Liouville problems with non-positive potentials, demonstrating that a specific a priori majorant is strictly less than ff.

Contribution

The paper introduces a precise a priori majorant for the first eigenvalues of Sturm--Liouville problems with constraints on the potential, proving it is strictly less than ff.

Findings

The inequality M_b<f^2 is proven.

The majorant M_b is shown to be strictly less than ^2.

The result applies to potentials q b 0 with integral constraints.

Abstract

Let $M_\gamma$ be precise a priori majorant of first eigenvalues of Sturm--Liouville problems $-y"+qy=\lambda y,\quad y(0)=y(1)=0$, where $q\leqslant 0$ and $\int_0^1 |q|^\gamma\,dx=1$, $\gamma\in (0,1/2)$. It is shown that the inequality $M_\gamma<\pi^2$ is true.