On majorants of eigenvalues of Sturm-Liouville problems with potentials from balls of weighted spaces

Abstract

It is constructively proved that for class $A_{r,\gamma}=\{q\in L_{1,loc}(0,1): q\leq 0, \int_0^1 rq^\gamma\,dx\leqslant 1\}$, where $r\in C[0,1]$ is uniformly positive weight and $\gamma>1$, there exists a unique potential $\hat q\in A_{r,\gamma}$ such that minimal eigenvalue $\lambda_0(\hat q)$ of boundary problem $$-y"+\hat qy=\lambda y, y(0)=y(1)=0 $$ is equal to $M_{r,\gamma}=\sup_{q\in A_{r,\gamma}}\lambda_0(q)$. For case $\gamma=1$ we obtain that there exists a unique potential $\hat q\in\Gamma_{r,\gamma}$ with analogous property. Here $\Gamma_{r,\gamma}$ is a closure of $A_{r,\gamma}$ in the space $W_{2,loc}^{-1}(0,1)$ of generalized functions.