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arXiv · 1412.7986

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

A. A. Vladimirov·E. S. Karulina
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Abstract

It is proved that for class $A_γ=\{q\in L_1[0,1]: q\geq 0, \int_0^1 q^γ\,dx=1\}$, where $γ\in (0,1)$, there exists a potential $q_*\in A_γ$ such that minimal eigenvalue $λ_1(q_*)$ of boundary problem $$ -y"+q_*y=λy, y'(0)=y'(1)=0 $$ is equal to $m_γ=\inf_{q\in A_γ}λ_1(q)$. The equality $m_γ=1$ for $γ\leq 1-2π^{-2}$ and the inequality $m_γ<1$ for $γ>1-2π^{-2}$ are also obtained.

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A. A. Vladimirov, E. S. Karulina. 2015-03-19. On a lower a priori estimate of minimal eigenvalue of one Sturm-Liouville problem with second-type boundary conditions. https://arxiv.org/abs/1412.7986

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