Geodesic
Spectral problem generated by the equation of smooth string with piece-wise constant friction
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 8 (2012) no. 3, pp. 280-295 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the paper, the spectral problem generated by the Sturm–Liouville equation $$ - y'' + q(x) y = (\lambda^2 - i p(x) \lambda) y, $$ where $q(x)$ is a real $L_2(0,a)$-function and $p(x)$ is a peace-wise constant, is considered with the Dirichlet boundary conditions at the ends of the interval $(0,a)$. The spectrum of the problem is compared with the spectra of auxiliary problems with the Dirichlet–Dirichlet and the Dirichlet–Neumann boundary conditions on the halves of the interval. Asymptotic formulas are obtained for the eigenvalues of this problem. 
@article{JMAG_2012_8_3_a4,
     author = {L. Kobyakova},
     title = {Spectral problem generated by the equation of smooth string with piece-wise constant friction},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {280--295},
     year = {2012},
     volume = {8},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2012_8_3_a4/}
}
TY  - JOUR
AU  - L. Kobyakova
TI  - Spectral problem generated by the equation of smooth string with piece-wise constant friction
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2012
SP  - 280
EP  - 295
VL  - 8
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/JMAG_2012_8_3_a4/
LA  - ru
ID  - JMAG_2012_8_3_a4
ER  - 
%0 Journal Article
%A L. Kobyakova
%T Spectral problem generated by the equation of smooth string with piece-wise constant friction
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2012
%P 280-295
%V 8
%N 3
%U http://geodesic.mathdoc.fr/item/JMAG_2012_8_3_a4/
%G ru
%F JMAG_2012_8_3_a4
L. Kobyakova. Spectral problem generated by the equation of smooth string with piece-wise constant friction. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 8 (2012) no. 3, pp. 280-295. http://geodesic.mathdoc.fr/item/JMAG_2012_8_3_a4/

[1] F.V. Atkinson, Discreet and Continuous Boundary Problems, Mir, Moscow, 1968 (Russian) | MR | Zbl

[2] I.S. Kats and M.G. Krein, “On the Spectral Functions of the String”, Appendix II: F.V. Atkinson, Discreet and Continuous Boundary Problems, Mir, Moscow, 1968 (Russian)

[3] R. Courant and D. Gilbert, Methods of Mathematical Physics, v. I, Interscience Publishers, Inc., New York, 1953

[4] Y.M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, Naukova dumka, Kiev, 1965 (Russian) | MR

[5] A.G. Kostuchenko and I.S. Sargsyan, Distribution of Eigenvalues (Selfadjoint Simple Differential Operator), Nauka, Moscow, 1979 (Russian) | MR

[6] J. Pöschl and E. Trubowitz, Inverse Scattering Theory, Academic Press, New York, 1987

[7] D.Z. Arov, “Realization of a Canonical System with a Dissipative Boundary Condition at One End of the Segment in Terms of the Coefficient of Dynamical Compliance”, Sib. Mat. Zh., 16:3 (1975), 440–463 (Russian) | MR | Zbl

[8] M.G. Krein and A.A. Nudelman, “On Direct and Inverse Problems for Boundary Dissipation Frequencies of a Nonuniform String”, Dokl. AN SSSR, 247:5 (1979), 1046–1049 (Russian) | MR

[9] M.G. Krein and A.A. Nudelman, “On Some Spectral Properties of a Non-Homogeneous String with a Dissipative Boundary Condition”, J. Oper. Theory, 22 (1989), 369–395 (Russian) | MR | Zbl

[10] V.N. Pivovarchik, “Inverse Problem for a Smooth String with Damping at One End”, J. Oper. Theory, 38 (1998), 243–263 | MR

[11] V.N. Pivovarchik, “Direct and Inverse Problems for a Damped String”, J. Oper. Theory, 42 (1999), 189–220 | MR | Zbl

[12] M. Jaulent, “Inverse Scattering Problems in Absorbing Media”, J. Math. Phys., 17 (1976), 1351–1360 | DOI | MR

[13] C. Jean and M. Jaulent, “The Inverse Problem for the One-Dimensional Schrödinger Equation with an Energy-Dependent polential”, Ann. Inst. Henri Poincaré, 25:2 (1976), 119–137 | MR | Zbl

[14] M.G. Gasymov and G.Sh. Guseynov, “Determination of a Diffusion Operator from Spectral Data”, Dokl. AN AzSSR, 37:2 (1981), 19–23 (Russian) | MR | Zbl

[15] Chuan-Fu Yang, “New Trace Formulae for a Quadratic Pencil of the Schrödinger Operator”, J. Math. Phys., 51 (2010), 033506 | DOI | MR

[16] S. Cox and E. Zuazua, “The Rate at which Energy Decays in a String Damped at the End”, Indiana Univ. Math. J., 44 (1995), 545–573 | DOI | MR | Zbl

[17] V.N. Pivovarchik, “On Spectra of a Certain Class of Quadratic Operator Pencils with One-Dimencional Linear Part”, Ukr. Mat. Zh., 59:5 (2007), 702–716 | DOI | MR | Zbl

[18] B.Ya. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Mono, 5, AMS, Providence, RI, 1980 | MR

[19] Birkhauser Verlag, 1986, 367 pp. | MR | Zbl