Higher derivatives of spectral functions associated with one-dimensional schrödinger operators

D. J. Gilbert; B. J. Harris; S. M. Riehl

doi:10.1007/978-3-7643-8755-6_10

Higher derivatives of spectral functions associated with one-dimensional schrödinger operators

D. J. Gilbert, B. J. Harris, S. M. Riehl

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We investigate the existence and asymptotic behaviour of higher derivatives of the spectral function, p(λ), on the positive real axis, 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 nth derivative, p ⁽ⁿ⁾ (λ), and outline a systematic procedure for estimating numerical upper bounds for the turning points of such derivatives. The potential relevance of our results to some topical issues in spectral theory is discussed.

Original language	English
Title of host publication	Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006
Editors	Jan Janas, Pavel Kurasov, Ari Laptev, Ari Laptev, Sergei Naboko, Gunter Stolz
Publisher	Springer International Publishing
Pages	217-228
Number of pages	12
ISBN (Print)	9783764387549
DOIs	https://doi.org/10.1007/978-3-7643-8755-6_10
Publication status	Published - 2009
Event	Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006 - Lund, Sweden Duration: 1 Jan 2006 → …

Publication series

Name	Operator Theory: Advances and Applications
Volume	186
ISSN (Print)	0255-0156
ISSN (Electronic)	2296-4878

Conference

Conference	Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006
Country/Territory	Sweden
City	Lund
Period	1/01/06 → …

Keywords

Spectral functions
Sturm-liouville problems
Unbounded selfadjoint operators

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10.1007/978-3-7643-8755-6_10

https://arrow.tudublin.ie/scschmatart/111Licence: CC BY-NC-SA

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Gilbert, D. J., Harris, B. J., & Riehl, S. M. (2009). Higher derivatives of spectral functions associated with one-dimensional schrödinger operators. In J. Janas, P. Kurasov, A. Laptev, A. Laptev, S. Naboko, & G. Stolz (Eds.), Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006 (pp. 217-228). (Operator Theory: Advances and Applications; Vol. 186). Springer International Publishing. https://doi.org/10.1007/978-3-7643-8755-6_10

Gilbert, D. J. ; Harris, B. J. ; Riehl, S. M. / Higher derivatives of spectral functions associated with one-dimensional schrödinger operators. Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006. editor / Jan Janas ; Pavel Kurasov ; Ari Laptev ; Ari Laptev ; Sergei Naboko ; Gunter Stolz. Springer International Publishing, 2009. pp. 217-228 (Operator Theory: Advances and Applications).

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title = "Higher derivatives of spectral functions associated with one-dimensional schr{\"o}dinger operators",

abstract = "We investigate the existence and asymptotic behaviour of higher derivatives of the spectral function, p(λ), on the positive real axis, in the context of one-dimensional Schr{\"o}dinger operators on the half-line with integrable potentials. In particular, we identify sufficient conditions on the potential for the existence and continuity of the nth derivative, p (n) (λ), and outline a systematic procedure for estimating numerical upper bounds for the turning points of such derivatives. The potential relevance of our results to some topical issues in spectral theory is discussed.",

keywords = "Spectral functions, Sturm-liouville problems, Unbounded selfadjoint operators",

author = "Gilbert, \{D. J.\} and Harris, \{B. J.\} and Riehl, \{S. M.\}",

note = "Publisher Copyright: {\textcopyright} 2008 Birkh{\"a}user Verlag Basel/Switzerland.; Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006 ; Conference date: 01-01-2006",

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editor = "Jan Janas and Pavel Kurasov and Ari Laptev and Ari Laptev and Sergei Naboko and Gunter Stolz",

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Gilbert, DJ, Harris, BJ & Riehl, SM 2009, Higher derivatives of spectral functions associated with one-dimensional schrödinger operators. in J Janas, P Kurasov, A Laptev, A Laptev, S Naboko & G Stolz (eds), Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006. Operator Theory: Advances and Applications, vol. 186, Springer International Publishing, pp. 217-228, Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006, Lund, Sweden, 1/01/06. https://doi.org/10.1007/978-3-7643-8755-6_10

Higher derivatives of spectral functions associated with one-dimensional schrödinger operators. / Gilbert, D. J.; Harris, B. J.; Riehl, S. M.
Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006. ed. / Jan Janas; Pavel Kurasov; Ari Laptev; Ari Laptev; Sergei Naboko; Gunter Stolz. Springer International Publishing, 2009. p. 217-228 (Operator Theory: Advances and Applications; Vol. 186).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - We investigate the existence and asymptotic behaviour of higher derivatives of the spectral function, p(λ), on the positive real axis, 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 nth derivative, p (n) (λ), and outline a systematic procedure for estimating numerical upper bounds for the turning points of such derivatives. The potential relevance of our results to some topical issues in spectral theory is discussed.

AB - We investigate the existence and asymptotic behaviour of higher derivatives of the spectral function, p(λ), on the positive real axis, 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 nth derivative, p (n) (λ), and outline a systematic procedure for estimating numerical upper bounds for the turning points of such derivatives. The potential relevance of our results to some topical issues in spectral theory is discussed.

KW - Spectral functions

KW - Sturm-liouville problems

KW - Unbounded selfadjoint operators

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DO - 10.1007/978-3-7643-8755-6_10

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SN - 9783764387549

T3 - Operator Theory: Advances and Applications

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BT - Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006

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A2 - Kurasov, Pavel

A2 - Laptev, Ari

A2 - Naboko, Sergei

A2 - Stolz, Gunter

PB - Springer International Publishing

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Gilbert DJ, Harris BJ, Riehl SM. Higher derivatives of spectral functions associated with one-dimensional schrödinger operators. In Janas J, Kurasov P, Laptev A, Laptev A, Naboko S, Stolz G, editors, Methods of Spectral Analysis in Mathematical Physics - Conference on Operator Theory, Analysis and Mathematical Physics, OTAMP 2006. Springer International Publishing. 2009. p. 217-228. (Operator Theory: Advances and Applications). doi: 10.1007/978-3-7643-8755-6_10

Higher derivatives of spectral functions associated with one-dimensional schrödinger operators

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