Bounds for the points of spectral concentration of Sturm–Liouville problems

Gilbert, Daphne; Harris, Bernard

doi:10.1112/s002557930001593x

Cited by 10 publications

(13 citation statements)

References 8 publications

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“…This question was also considered in [2] where the physical interpretation of such points was discussed. The results of [6] lead to a computable L 1 which is such that r 0 ðlÞ has no points of spectral concentration for lXL 1 :…”

Section: Introductionmentioning

confidence: 99%

“…It is known that if qAL 1 ½0; NÞ; then the spectrum is purely absolutely continuous on (0; NÞ; r 0 a ðlÞ exists, is continuous in l and satisfies r 0 a ðlÞ40 for l40 (see for example [8,15]). In [9] a series representation was given for r 0 a ðlÞ for l4L 0 where L 0 is computable under general conditions which require little more than qAL 1 ½0; NÞ: In [6] the question of spectral concentration was also considered under the same circumstances. In this case, points of spectral concentration are defined, roughly, as values of lAð0; NÞ at which r 0 a ðlÞ has a local maximum.…”

Section: Introductionmentioning

confidence: 99%

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The spectral function for Sturm–Liouville problems where the potential is of Wigner–von Neumann type or slowly decaying

Gilbert¹,

Harris

Riehl

2004

Journal of Differential Equations

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We consider the linear, second-order, differential equationwith the boundary condition yð0Þ cos a þ y 0 ð0Þ sin a ¼ 0 for some aA½0; pÞ: ðÃÃÞWe suppose that qðxÞ is real-valued, continuously differentiable and that qðxÞ-0 as x-N with qeL 1 ½0; NÞ: Our main object of study is the spectral function r a ðlÞ associated with (Ã) and (ÃÃ). We derive a series expansion for this function, valid for lXL 0 where L 0 is computable and establish a L 1 ; also computable, such that (Ã) and (ÃÃ) with a ¼ 0; have no points of spectral concentration for lXL 1 : We illustrate our results with examples. In particular we consider the case of the Wigner-von Neumann potential.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The spectral function for Sturm–Liouville problems where the potential is of Wigner–von Neumann type or slowly decaying

Gilbert¹,

Harris

Riehl

2004

Journal of Differential Equations

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“…In the case of the Schrödinger operator (1.1) considered in, for example, [1,3,7,8,11,24, § § 5.7 and 5.10], an integral formula for the spectral density is derived as a consequence of the asymptotic form of the solutions of the equation ly = λy as x → ∞ (see [6]). Here λ is the complex spectral parameter.…”

Section: Asymptotics Of Solutions and The Spectral Densitymentioning

confidence: 99%

Absence of high-energy spectral concentration for Dirac systems with divergent potentials

Eastham

Schmidt

2005

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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It is known that one-dimensional Dirac systems with potentials q which tend to −∞ (or ∞) at infinity, such that 1/q is of bounded variation, have a purely absolutely continuous spectrum covering the whole real line. We show that, for the system on a half-line, there are no local maxima of the spectral density (points of spectral concentration) above some value of the spectral parameter if q satisfies certain additional regularity conditions. These conditions admit thrice-differentiable potentials of power or exponential growth. The eventual sign of the derivative of the spectral density depends on the boundary condition imposed at the regular end-point.

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“…It may be shown that the Riccati equation has at most one solution of this form (see [7]), from which it follows that if the series representation in (7) is a valid solution of (4), then it does indeed represent the extension of the generalised Dirichlet m-function onto the real axis, as sought. Substituting for m(x, λ) from (7) into (4) and rearranging yields…”

Section: Short Range Potentialsmentioning

confidence: 99%

“…The following lemma establishes some key properties of {m n (x, λ)} and {w n (x, λ)}, and is proved in [7].…”

Section: Short Range Potentialsmentioning

confidence: 99%

Higher Derivatives of Spectral Functions Associated with One-Dimensional Schrödinger Operators

Gilbert¹,

Harris

Riehl

Methods of Spectral Analysis in Mathematical Physics

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We investigate the existence and asymptotic behaviour of higher derivatives of the spectral function in the context of one-dimensional Schrödinger operators on the half-line with integrable potentials. In particular, we identify sufficient conditions on the potential for the existence and continuity of the n-th derivative, and outline a systematic procedure for estimating numerical upper bounds for the turning points of such derivatives. Explicit worked examples illustrate the development and application of the theory.

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Bounds for the points of spectral concentration of Sturm–Liouville problems

Cited by 10 publications

References 8 publications

The spectral function for Sturm–Liouville problems where the potential is of Wigner–von Neumann type or slowly decaying

The spectral function for Sturm–Liouville problems where the potential is of Wigner–von Neumann type or slowly decaying

Absence of high-energy spectral concentration for Dirac systems with divergent potentials

Higher Derivatives of Spectral Functions Associated with One-Dimensional Schrödinger Operators

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