Sturm-Liouville operators with singular potentials

Савчук, А. М.

doi:10.1007/s10958-007-0268-1

Cited by 79 publications

(191 citation statements)

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“…In the recent work by Shkalikov and Savchuk (see [25,26] and the references therein) most of the classical Sturm-Liouville theory has been generalized to the case of distributional potentials in W −1 2 (0, 1). The corresponding self-adjoint operators could be defined by the form sum method; however, such a definition is rather abstract and does not take into account the differential nature of these operators.…”

Section: )mentioning

confidence: 99%

“…The corresponding self-adjoint operators could be defined by the form sum method; however, such a definition is rather abstract and does not take into account the differential nature of these operators. The explicit construction of [25,26] define the same operators as differential ones; it rests on the regularization by quasi-derivatives and proceeds as follows. Let q be a real-valued distribution from W −1 2 (0, 1) and let σ ∈ L 2 (0, 1) be any of its distributional primitives; then the differential expression (1.1) (understood in the sense of distributions) can be regularized through…”

Section: )mentioning

confidence: 99%

“…It is known [1,11,26] that the sequences (λ n ) and (µ n ) satisfy (A1) and the following relaxed version of (A2):…”

Section: )mentioning

confidence: 99%

“…For the sake of definiteness, we shall concentrate on the case h 0 = h 1 = ∞ and h 0 = h ∈ R only, the other cases being analogous. More precisely, we shall formulate necessary and sufficient conditions on sequences (λ n ) and (µ n ) in order that they should be eigenvalues of the Sturm-Liouville operators T (q, ∞, ∞) and T (q, h, ∞), respectively, for some choice of q in W s−1 2 (0, 1) and h ∈ R. For an arbitrary intermediate value s ∈ (0, 1), the direct spectral problem was studied in [10,13,26]. For instance, it was proved in [10] that the eigenvalue remaindersλ n andμ n defined in (1.5) are, respectively, even and odd sine Fourier coefficients of some function from W s 2 (0, 1) (cf.…”

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confidence: 99%

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Inverse Spectral Problems for Sturm–liouville Operators With Singular Potentials. Iv. Potentials in the Sobolev Space Scale

Hryniv¹,

Mykytyuk²

2006

Proceedings of the Edinburgh Mathematical Society

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Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

“…It is known [1,11,26] that the sequences (λ n ) and (µ n ) satisfy (A1) and the following relaxed version of (A2):…”

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

See 2 more Smart Citations

Inverse Spectral Problems for Sturm–liouville Operators With Singular Potentials. Iv. Potentials in the Sobolev Space Scale

Hryniv¹,

Mykytyuk²

2006

Proceedings of the Edinburgh Mathematical Society

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“…In work [5], a well-defined Sturm-Liouville operator = − ′′ ( ) + ′ ( ) ( ) is provided and there was proved the existence of the limit of its resolvent in the case → ∈ 2 (0, 1). In work [6] these results were extended for linear operators of higher even order.…”

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confidence: 99%

Perturbation of second order nonlinear equation by delta-like potential

Gadyl’shin¹,

Mukminov²

2018

Ufimskii Matematicheskii Zhurnal

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Abstract. We consider boundary value problems for one-dimensional second order quasilinear equation on bounded and unbounded intervals of the real axis. The equation perturbed by the delta-shaped potential −1 (︀ −1 )︀ , where ( ) is a compactly supported function, 0 < ≪ 1. The mean value of ⟨ ⟩ can be negative, but it is assumed to be bounded from below ⟨ ⟩ − 0 . The number 0 is defined in terms of coefficients of the equation. We study the convergence rate of the solution of the perturbed problem to the solution of the limit problem 0 as the parameter tends to zero. In the case of a bounded interval , the estimate of the form | ( ) − 0 ( )| < is established. As the interval is unbounded, we prove a weaker estimate | ( ) − 0 ( )/ < 1/2 . The estimates are proved by using original cut-off functions as trial functions. For simplicity, the proof of the existence of solutions to perturbed and limiting problems are made by the method of contracting mappings. The disadvantage of this approach, as it is known, is the smallness of the nonlinearities in the equation. We consider the cases of the Dirichlet, Neumann and Robin condition.

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On spectral theory for Schrödinger operators with strongly singular potentials

Gesztesy¹,

Zinchenko

2006

Mathematische Nachrichten

168

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Abstract. We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schrödinger operators on [a, ∞), a ∈ R, with a regular finite end point a and the case of Schrödinger operators on the real line with locally integrable potentials, which naturally lead to Herglotz functions and 2 × 2 matrix-valued Herglotz functions representing the associated Weyl-Titchmarsh coefficients. Second, we contrast this with the case of self-adjoint half-line Schrödinger operators on (a, ∞) with a potential strongly singular at the end point a. We focus on situations where the potential is so singular that the associated maximally defined Schrödinger operator is selfadjoint (equivalently, the associated minimally defined Schrödinger operator is essentially self-adjoint) and hence no boundary condition is required at the finite end point a. For this case we show that the Weyl-Titchmarsh coefficient in this strongly singular context still determines the associated spectral function, but ceases to posses the Herglotz property. However, as will be shown, Herglotz function techniques continue to play a decisive role in the spectral theory for strongly singular Schrödinger operators.

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Sturm-Liouville operators with singular potentials

Cited by 79 publications

References 7 publications

Inverse Spectral Problems for Sturm–liouville Operators With Singular Potentials. Iv. Potentials in the Sobolev Space Scale

Inverse Spectral Problems for Sturm–liouville Operators With Singular Potentials. Iv. Potentials in the Sobolev Space Scale

Perturbation of second order nonlinear equation by delta-like potential

On spectral theory for Schrödinger operators with strongly singular potentials

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