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Povzner–Wienholtz-Type Theorems for Sturm–Liouville Operators with Singular Coefficients

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Abstract

We introduce and investigate symmetric operators \(L_0\) associated in the complex Hilbert space \(L^2({\mathbb {R}})\) with a formal differential expression

$$\begin{aligned} l[u] :=-(pu')'+qu + i((ru)'+ru') \end{aligned}$$

under minimal conditions on the regularity of the coefficients. They are assumed to satisfy conditions

$$\begin{aligned} q=Q'+s;\quad \frac{1}{\sqrt{|p|}}, \frac{Q}{\sqrt{|p|}}, \frac{r}{\sqrt{|p|}} \in L^2_{loc}\left( {\mathbb {R}}\right) , \quad s \in L^1_{loc}\left( {\mathbb {R}}\right) , \end{aligned}$$

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 \({\mathbb {R}}\), while function p may be discontinuous. The main result of the paper are two sufficient conditions on the coefficient p which provide that the operator \(L_0\) being semi-bounded implies it being self-adjoint.

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Authors and Affiliations

  1. Institute of Mathematics of the Czech Academy of Sciences, Zitna str., 25, 11567, Prague, Czech Republic

    Vladimir Mikhailets

  2. Institute of Mathematics of NAS of Ukraine, Tereshchenkivska str., 3, Kyiv, 01024, Ukraine

    Vladimir Mikhailets, Andrii Goriunov & Volodymyr Molyboga

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  1. Vladimir Mikhailets
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  2. Andrii Goriunov
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  3. Volodymyr Molyboga
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Correspondence to Vladimir Mikhailets.

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Communicated by Gerald Teschl.

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Mikhailets, V., Goriunov, A. & Molyboga, V. Povzner–Wienholtz-Type Theorems for Sturm–Liouville Operators with Singular Coefficients. Complex Anal. Oper. Theory 16, 113 (2022). https://doi.org/10.1007/s11785-022-01291-y

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