N. E. Firsova, “An inverse scattering problem for a perturbed Hill's operator”, Mat. Zametki, 18:6 (1975), 831–843; Math. Notes, 18:6 (1975), 1085

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Matematicheskie Zametki, 1975, Volume 18, Issue 6, Pages 831–843 (Mi mzm7695)

This article is cited in 17 scientific papers (total in 17 papers)

An inverse scattering problem for a perturbed Hill's operator

N. E. Firsova

Leningrad State University

Full-text PDF (910 kB) Citations (17)

Abstract: We consider the inverse scattering problem for the operator

L = - d^{2} / d x^{2} + p (x) + q (x)

$L=-d^2/dx^2+p(x)+q(x)$ ,

x \in R^{1}

$x\in R^1$ . The perturbation potential

q

$q$ is expressed in terms of the periodic potential

p

$p$ and the scattering data. We also obtain identities for the eigenfunctions of the unperturbed Hill's operator

L_{0} = - d^{2} / d x^{2} + p (x)

$L_0=-d^2/dx^2+p(x)$ .

Received: 19.11.1974

English version:
Mathematical Notes, 1975, Volume 18, Issue 6, Pages 1085–1091
DOI: https://doi.org/10.1007/BF01099986

Bibliographic databases:

UDC: 517

Language: Russian

Citation: N. E. Firsova, “An inverse scattering problem for a perturbed Hill's operator”, Mat. Zametki, 18:6 (1975), 831–843; Math. Notes, 18:6 (1975), 1085–1091

Citation in format AMSBIB

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\by N.~E.~Firsova

\paper An inverse scattering problem for a~perturbed Hill's operator

\jour Mat. Zametki

\yr 1975

\vol 18

\issue 6

\pages 831--843

\mathnet{http://mi.mathnet.ru/mzm7695}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=407364}

\zmath{https://zbmath.org/?q=an:0321.34016}

\transl

\jour Math. Notes

\yr 1975

\vol 18

\issue 6

\pages 1085--1091

\crossref{https://doi.org/10.1007/BF01099986}

Linking options:

https://www.mathnet.ru/eng/mzm7695

https://www.mathnet.ru/eng/mzm/v18/i6/p831

This publication is cited in the following 17 articles:

A. Kh. Khanmamedov, M. F. Muradov, “Operator preobrazovaniya dlya uravneniya Shredingera s dopolnitelnym eksponentsialnym potentsialom”, Izv. vuzov. Matem., 2023, no. 9, 76–84
A. Kh. Khanmamedov, D. H. Orudjov, “On transformation operators for the Schrödinger equation with an additional periodic complex potential”, Bol. Soc. Mat. Mex., 29:2 (2023)
A. Kh. Khanmamedov, M. F. Muradov, “Transformation Operator for the Schrödinger Equation with Additional Exponential Potential”, Russ Math., 67:9 (2023), 68
A. Kh. Khanmamedov, A. F. Mamedova, “A Remark on the Inverse Scattering Problem for the Perturbed Hill Equation”, Math. Notes, 112:2 (2022), 281–285
A. R. Bikmetov, I. Kh. Khusnullin, “Perturbation of Hill operator by narrow potentials”, Russian Math. (Iz. VUZ), 61:7 (2017), 1–10
I. Egorova, Z. Gladka, T. L. Lange, G. Teschl, “Inverse Scattering Theory for Schrödinger Operators with Steplike Potentials”, Zhurn. matem. fiz., anal., geom., 11:2 (2015), 123–158
Alice Mikikits-Leitner, Gerald Teschl, Spectral Theory and Analysis, 2011, 107
I. Egorova, G. Teschl, “A Paley–Wiener theorem for periodic scattering with applications to the Korteweg–de Vries equation”, Zhurn. matem. fiz., anal., geom., 6:1 (2010), 21–33
S. V. Frolov, “A spectral identity for the multidimensional Schrödinger equation with periodic potential”, Theoret. and Math. Phys., 91:3 (1992), 692–695
Thomas M. Roberts, “Introduction to Schrödinger inverse scattering”, Physica B: Condensed Matter, 173:1-2 (1991), 157
T M Roberts, “Inverse scattering for step-periodic potentials in one dimension”, Inverse Problems, 6:5 (1990), 797
Thomas M. Roberts, “Scattering for step-periodic potentials in one dimension”, Journal of Mathematical Physics, 31:9 (1990), 2181
N. E. Firsova, “On solution of the Cauchy problem for the Korteweg–de Vries equation with initial data the sum of a periodic and a rapidly decreasing function”, Math. USSR-Sb., 63:1 (1989), 257–265
N. E. Firsova, “The direct and inverse scattering problems for the one-dimensional perturbed Hill operator”, Math. USSR-Sb., 58:2 (1987), 351–388
N. E. Firsova, “Levinson formula for perturbed Hill operator”, Theoret. and Math. Phys., 62:2 (1985), 130–140
P. A. Kuchment, “Floquet theory for partial differential equations”, Russian Math. Surveys, 37:4 (1982), 1–60
N. E. Firsova, “Some spectral identities for the one-dimensional hill operator”, Theoret. and Math. Phys., 37:2 (1978), 1022–1027

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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