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

Abstract

It is constructively proved that for class $A_{r,\gamma }=\{q\in L_{1,loc}(0,1):q\le 0,{\int }_0^{1}r{q}^{\gamma }dx⩽1\}$, 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 $$-y"+\hat{q}y=\lambda y,y(0)=y(1)=0$$ 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.

15 pages, in Russian