Geodesic
One Inverse Problem for the Sturm--Liouville Operator
Matematičeskie zametki, Tome 110 (2021) no. 1, pp. 3-16.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the spectral problems for the Sturm–Liouville operator generated by the Dirichlet, Neumann, Dirichlet–Neumann and Neumann–Dirichlet conditions. The necessary and sufficient condition for the coincidence of the spectrum of the Dirichlet-Neumann and Neumann–Dirichlet problems is obtained. The necessary and sufficient condition for the coincidence of the spectrum, except zero, of the Dirichlet and Neumann problems is also obtained. Their application to periodic and anti-periodic problems is given.
Keywords: Dirichlet problem, Neumann problem, Sturm–Liouville operator, coincidence of the spectrum. 
@article{MZM_2021_110_1_a0,
     author = {B. N. Biyarov},
     title = {One {Inverse} {Problem} for the {Sturm--Liouville} {Operator}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {3--16},
     publisher = {mathdoc},
     volume = {110},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a0/}
}
TY  - JOUR
AU  - B. N. Biyarov
TI  - One Inverse Problem for the Sturm--Liouville Operator
JO  - Matematičeskie zametki
PY  - 2021
SP  - 3
EP  - 16
VL  - 110
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a0/
LA  - ru
ID  - MZM_2021_110_1_a0
ER  - 
%0 Journal Article
%A B. N. Biyarov
%T One Inverse Problem for the Sturm--Liouville Operator
%J Matematičeskie zametki
%D 2021
%P 3-16
%V 110
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a0/
%G ru
%F MZM_2021_110_1_a0
B. N. Biyarov. One Inverse Problem for the Sturm--Liouville Operator. Matematičeskie zametki, Tome 110 (2021) no. 1, pp. 3-16. http://geodesic.mathdoc.fr/item/MZM_2021_110_1_a0/

[1] G. Borg, “Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte”, Acta Math., 78 (1946), 1–96 | DOI | MR

[2] H. Hochstadt, “On the determination of a Hill's equation from its spectrum”, Arch. Rational Mech. Anal., 19 (1965), 353–362 | DOI | MR

[3] H. Hochstadt, “A direct and inverse problem for a Hill's equation with double eigenvalues”, J. Math. Anal. Appl., 66 (1978), 507–513 | DOI | MR

[4] H. Hochstadt, B. Lieberman, “An inverse Sturm–Liouville problem with mixed given data”, SIAM J. Appl. Math., 34:4 (1978), 676–680 | DOI | MR

[5] F. Gesztesy, B. Simon, “nverse spectral analysis with partial information on the potential. I. The case of an a.c. component in the spectrum”, Helv. Phys. Acta., 70:1-2 (1997), 66–71 | MR

[6] F. Gesztesy, B. Simon, “Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum”, II. The case of discrete spectrum, Trans. Amer. Math. Soc., 452:6 (1999), 2765–2787 | DOI | MR

[7] O. A. Veliev, A. A. Shkalikov, “O bazisnosti Rissa sobstvennykh i prisoedinennykh funktsii periodicheskoi i antiperiodicheskoi zadach Shturma–Liuvillya”, Matem. zametki, 85:5 (2009), 671–686 | DOI | MR | Zbl

[8] B. M. Levitan, Obratnye zadachi Shturma–Liuvillya, Nauka, M., 1988 | MR

[9] A. M. Savchuk, A. A. Shkalikov, “Obratnye zadachi dlya operatora Shturma–Liuvillya s potentsialami iz prostranstv Soboleva. Ravnomernaya ustoichivost”, Funkts. analiz i ego pril., 44:4 (2010), 34–53 | DOI | MR | Zbl

[10] V. A. Yurko, Vvedenie v teoriyu obratnykh spektralnykh zadach, Fizmatlit, M., 2006

[11] V. A. Marchenko, Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR