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Sturm-Liouville operators with distribution potentials. (English. Russian original) Zbl 1066.34085

Trans. Mosc. Math. Soc. 2003, 143-192 (2003); translation from Tr. Mosk. Mat. O.-va 64, 159-212 (2003).
Summary: This paper presents four different approaches to the definition of the Sturm-Liouville operator \(Ly= -y''+ q(x)y\) on an interval \((a, b)\) in the case when the potential \(q(x)\) is a distribution in a Sobolev space with negative index of smoothness, namely, \(q\in W^{-\theta}_2\) for \(\theta\leq 1\). The principal and secondary terms of the asymptotics are obtained for the eigenvalues and eigenfunctions of the operators defined, along with estimates on the remainders in these asymptotic formulas, in dependence on the class of the potential. The corresponding operators are defined and studied also for certain analytic families of potentials of high singularity that do not belong to the space \(W^{-1}_2\).

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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