Asymptotic integration of symmetric second-order quasidifferential equations

Konechnaya, N. N.

doi:10.1134/s0001434611110241

Cited by 2 publications

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“…In particular, the proof of (iv) is based on the recent improvement by Further notes: A generalization and further developments of Example 2.11 can be found in [64]. Using the approach based on quasi-derivatives, it was noticed in [58] and [84], [68] that the analysis of [56], [57] and [83] extends to the case of Hamiltonians with δ-interactions. In particular, using this approach one can extend Corollary 2.10 to the case of semibounded potentials q, q(.)…”

Section: 3mentioning

confidence: 99%

1-D Schrödinger operators with local point interactions: a review

Kostenko¹,

Malamud²

2013

Preprint

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Section: 3mentioning

confidence: 99%

1-D Schrödinger operators with local point interactions: a review

Kostenko¹,

Malamud²

2013

Preprint

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“…We do not dwell upon generalizations and applications of Theorems 10 and 11, in particular those to differential expressions of the form (8), and refer the reader to Konechnaya's paper [4].…”

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confidence: 99%

“…. Later, quite a few various examples of this type were constructed in [4] and [5]. If the function q(x) satisfies some of conditions (a)-(d) in Theorem 8, say, on the intervals In our situation, the expression l[f ] of the form (8) is considered, and the conditions given above ensure that the limit point case is realized for this expression.…”

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confidence: 99%

Sturm–Liouville operators

Mirzoev

2014

Trans. Moscow Math. Soc.

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Let (a, b) ⊂ R be a finite or infinite interval, let p 0 (x), q 0 (x), and p 1 (x), x ∈ (a, b), be real-valued measurable functions such that p 0 , p −1 0 , p 2 1 p −1 0 , and q 2 0 p −1 0 are locally Lebesgue integrable (i.e., lie in the space L 1 loc (a, b)), and let w(x), x ∈ (a, b), be an almost everywhere positive function. This paper gives an introduction to the spectral theory of operators generated in the space L 2 w (a, b) by formal expressions of the formwhere all derivatives are understood in the sense of distributions. The construction described in the paper permits one to give a sound definition of the minimal operator L 0 generated by the expression l[f ] in L 2 w (a, b) and include L 0 in the class of operators generated by symmetric (formally self-adjoint) second-order quasi-differential expressions with locally integrable coefficients. In what follows, we refer to these operators as Sturm-Liouville operators. Thus, the well-developed spectral theory of second-order quasi-differential operators is used to study Sturm-Liouville operators with distributional coefficients. The main aim of the paper is to construct a Titchmarsh-Weyl theory for these operators. Here the problem on the deficiency indices of L 0 , i.e., on the conditions on p 0 , q 0 , and p 1 under which Weyl's limit point or limit circle case is realized, is a key problem. We verify the efficiency of our results for the example of a Hamiltonian with δ-interactions of intensities h k centered at some points x k , where2010 Mathematics Subject Classification. Primary 34B24; Secondary 34B20, 34B40. Key words and phrases. Second-order quasi-differential operator, minimal operator, maximal operator, Sturm-Liouville theory, deficiency index, limit point-limit circle, linear differential equation with distributional coefficients, finite-difference equation, Jacobi matrix.

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Asymptotic integration of symmetric second-order quasidifferential equations

Cited by 2 publications

References 1 publication

1-D Schrödinger operators with local point interactions: a review

1-D Schrödinger operators with local point interactions: a review

Sturm–Liouville operators

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