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

2014/12/26 by A. A. Vladimirov, Vladimirov, A. A.
Mathematics · #03F60 #34L15 #FOS: Mathematics #Spectral Theory (math.SP) #math.SP #msc:03F60 #msc:34L15

paper · pdf · doi:10.48550/arxiv.1412.7992

15 pages, in Russian

arxiv created 2015/03/19 · arxiv updated 2015/03/20

Abstract

It is constructively proved that for class A_r,γ=\q∈ L_1,loc(0,1): q≤ 0, ∫₀¹ rq^γ dx\leqslant 1\, where r∈ C[0,1] is uniformly positive weight and γ>1, there exists a unique potential q∈ A_r,γ such that minimal eigenvalue λ₀( q) of boundary problem -y"+ qy=λy, y(0)=y(1)=0 is equal to M_r,γ=sup_q∈ A_r,γλ₀(q). For case γ=1 we obtain that there exists a unique potential q∈Γ_r,γ with analogous property. Here Γ_r,γ is a closure of A_r,γ in the space W_2,loc^-1(0,1) of generalized functions.

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