2021/10/22 by Andrii Goriunov, Goriunov, Andrii, Vladimir Mikhailets +3 · 2 citations
Mathematics · #Spectral Theory in Mathematical Physics #Differential Equations and Boundary Problems #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2110.11750
We introduce and investigate symmetric operators L0 associated in the complex Hilbert space L2(ℝ) with a formal differential expression l[u] :=-(pu')'+qu + i((ru)'+ru') under minimal conditions on the regularity of the coefficients. They are assumed to satisfy conditions q=Q'+s; (1)/(√(|p|)), (Q)/(√(|p|)), (r)/(√(|p|)) ∈ L2loc(ℝ), s ∈ L1loc(ℝ), (1)/(p)≠ 0 a.e., where the derivative of the function Q is understood in the sense of distributions, and all functions p, Q, r, s are real-valued. In particular, the coefficients q and r' may be Radon measures on ℝ, while function p may be discontinuous. The main result of the paper are constructive sufficient conditions on the coefficient p which provide that the operator L0 being semi-bounded implies it being self-adjoint.