On a lower a priori estimate of minimal eigenvalue of one Sturm-Liouville problem with second-type boundary conditions

Abstract

It is proved that for class $A_{\gamma }=\{q\in L_1[0,1]:q\ge 0,{\int }_0^1{q}^{\gamma }dx=1\}$, where $\gamma \in (0,1)$, there exists a potential $q_{\ast }\in A_{\gamma }$ such that minimal eigenvalue ${\lambda }_1(q_{\ast })$ of boundary problem $$-y"+q_{\ast }y=\lambda y,y^{\prime }(0)=y^{\prime }(1)=0$$ is equal to $m_{\gamma }={\inf }_{q\in A_{\gamma }}{\lambda }_1(q)$. The equality $m_{\gamma }=1$ for $\gamma \le 1-2{\pi }^{-2}$ and the inequality $m_{\gamma }<1$ for $\gamma >1-2{\pi }^{-2}$ are also obtained.

8 pages, in Russian