On an optimal potential of Schrödinger operator with prescribed m eigenvalue

2019/08/21 by Ilyasov, Yavdat, Valeev, Nurmukhamet
#34B15 #65R32 #74J25 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1908.07876

Abstract

The purpose of this paper is twofold: firstly, we present a new type of relationship between inverse problems and nonlinear differential equations. Secondly, we introduce a new type of inverse spectral problem, posed as follows: for a priori given potential V₀ find the closest function V such that m eigenvalues of one-dimensional space Schrodinger operator with potential V would coincide with the given values E₁ , … , E_m ∈ \mathbb R . In our main result, we prove the existence of a solution to this problem, and more importantly, we show that such a solution can be directly found by solving a system of nonlinear differential equations.

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