2025/12/02 by Vladimir Mikhailets, Mikhailets, Vladimir, Volodymyr Molyboga +1
Mathematics · #Differential Equations and Boundary Problems #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.SP #msc:34B20 #msc:34B24 #msc:34L40
paper · pdf · doi:10.48550/arxiv.2512.03215
19 pages, extended version
arxiv created 2026/07/29 · arxiv updated 2026/07/30
This paper studies the uniqueness problem for the one-dimensional Schrödinger operator associated with the formal differential expression l[u] =-u''+qu + i[(ru)'+ru'], in the complex Hilbert space L2(ℝ). The coefficients of the expression are complex-valued and satisfy q=s+Q', s ∈ L1loc(ℝ) \quadand Q, r ∈ L2loc(ℝ), where the derivative is understood in the sense of distributions. In particular, the potential q can be a Radon measure on the line. With the help of specially selected quasi-derivatives, the expression l is treated as a quasi-differential expression. The domains of the minimal L0 and maximal L operators associated with the expression l in the space L2(ℝ) are described. We find constructive conditions on the behaviour of Im r near ± ∞ that guarantee that L0=L if the operator L0 is accretive. We prove that these conditions are sharp even in the class of differential operators with smooth real-valued coefficients. Examples are given to illustrate the main results of the paper.