Geodesic
Invariant transformations for the Sturm–Liouville operator
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 25-54 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this paper we consider the Sturm–Liouville operator on a finite interval. For particular boundary conditions, a group of invariant transformations that preserve the operator's spectrum is constructed. This result permits us to reconsider some old problems for the Sturm–Liouville operator. In particular, the influence of this group on the inverse problem is discussed. 
@article{ZNSL_2005_327_a2,
     author = {A. Vagharshakyan},
     title = {Invariant transformations for the {Sturm{\textendash}Liouville} operator},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {25--54},
     year = {2005},
     volume = {327},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a2/}
}
TY  - JOUR
AU  - A. Vagharshakyan
TI  - Invariant transformations for the Sturm–Liouville operator
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2005
SP  - 25
EP  - 54
VL  - 327
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a2/
LA  - en
ID  - ZNSL_2005_327_a2
ER  - 
%0 Journal Article
%A A. Vagharshakyan
%T Invariant transformations for the Sturm–Liouville operator
%J Zapiski Nauchnykh Seminarov POMI
%D 2005
%P 25-54
%V 327
%U http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a2/
%G en
%F ZNSL_2005_327_a2
A. Vagharshakyan. Invariant transformations for the Sturm–Liouville operator. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 33, Tome 327 (2005), pp. 25-54. http://geodesic.mathdoc.fr/item/ZNSL_2005_327_a2/

[1] M. G. Krein, I. S. Kats, “Kriterii diskretnosti spektra singulyarnoi struny”, Izv. VUZóv, 1958, no. 2(3), 136–153 | MR

[2] M. G. Krein, “Reshenie obratnoi zadachi Shturma–Liuvillya”, Doklady AN SSSR, 76:1 (1951), 21–24 | MR | Zbl

[3] E. L. Isaacson, E. Trubowitz, “The inverse Sturm–Liouville problem, 1”, Comm. Pure Appl. Mat., 26 (1983), 767–783 | DOI | MR

[4] J. Poschel, E. Trubowitz, Inverse Spectral Theory, Pure Appl. Math., 130, Academic Press, 1987 | MR

[5] A. Vagharshakyan, On the spectrum of the Sturm–Liouville operator, Preprint TRITA-MAT-2000-09 (2000) KTH Stockholm

[6] V. Maz'a, T. Shaposhnikova, Multipliers in spaces of differentiable functions, St.Peterburg Univ., 1986 | MR

[7] V. A. Marchenko, Operatory Shturma–Liuvillya i ikh prilozheniya, Kiev, 1977 | MR | Zbl

[8] B. Sz.-Nagy, “Vibrations d'une corde nonhomogene”, Bull. Soc. Math. France, 75 (1947), 193–208 | MR

[9] Ph. Hartman, Ordinary differential equations, John Wiley Sons, New-York, 1964 | MR | Zbl

[10] Mu-Fa Chen, “Ergodic convergence rates of Markov processes – eigenvalues inequalities and ergodic theory”, ICM, 3 (2002), 41–52 | MR

[11] N. Danford, J. Schwartz, Linear operators, V. 2, New York, London, 1963 | MR

[12] A. Vagharshakyan, “A group of invariant transforms for Sturm–Liouville operators”, Algebra, Geometry, Appl., 2 (2002), 38–47 | MR | Zbl

[13] H. P. W. Gottlieb, “Isospectral strings”, Inverse Problems, 18 (2002), 971–978 | DOI | MR | Zbl