Geodesic
A Mapping Method in Inverse Sturm–Liouville Problems with Singular Potentials
Informatics and Automation, Differential equations and dynamical systems, Tome 261 (2008), pp. 243-248 
A. M. Savchuk. A Mapping Method in Inverse Sturm–Liouville Problems with Singular Potentials. Informatics and Automation, Differential equations and dynamical systems, Tome 261 (2008), pp. 243-248. http://geodesic.mathdoc.fr/item/TRSPY_2008_261_a17/
@article{TRSPY_2008_261_a17,
     author = {A. M. Savchuk},
     title = {A {Mapping} {Method} in {Inverse} {Sturm{\textendash}Liouville} {Problems} with {Singular} {Potentials}},
     journal = {Informatics and Automation},
     pages = {243--248},
     year = {2008},
     volume = {261},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2008_261_a17/}
}
TY  - JOUR
AU  - A. M. Savchuk
TI  - A Mapping Method in Inverse Sturm–Liouville Problems with Singular Potentials
JO  - Informatics and Automation
PY  - 2008
SP  - 243
EP  - 248
VL  - 261
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2008_261_a17/
LA  - ru
ID  - TRSPY_2008_261_a17
ER  - 
%0 Journal Article
%A A. M. Savchuk
%T A Mapping Method in Inverse Sturm–Liouville Problems with Singular Potentials
%J Informatics and Automation
%D 2008
%P 243-248
%V 261
%U http://geodesic.mathdoc.fr/item/TRSPY_2008_261_a17/
%G ru
%F TRSPY_2008_261_a17

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

In the space $L_2[0,\pi]$, the Sturm–Liouville operator $L_\mathrm D(y)=-y''+q(x)y$ with the Dirichlet boundary conditions $y(0)=y(\pi)=0$ is analyzed. The potential $q$ is assumed to be singular; namely, $q=\sigma'$, where $\sigma\in L_2[0,\pi]$, i.e., $q\in W_2^{-1}[0,\pi]$. The inverse problem of reconstructing the function $\sigma$ from the spectrum of the operator $L_\mathrm D$ is solved in the subspace of odd real functions $\sigma(\pi/2-x)=-\sigma(\pi/2+x)$. The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.

[1] Borg G., “Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe”, Acta math., 78 (1946), 1–96 | DOI | MR | Zbl

[2] Hryniv R. O., Mykytyuk Ya. V., “Inverse spectral problems for Sturm–Liouville operators with singular potentials. III: Reconstruction by three spectra”, J. Math. Anal. and Appl., 284:2 (2003), 626–646 | DOI | MR | Zbl

[3] Levitan B. M., Obratnye zadachi Shturma–Liuvillya, Nauka, M., 1984 | MR | Zbl

[4] McLaughlin J. R., “Inverse spectral theory using nodal points as data – A uniquenes result”, J. Diff. Equat., 73 (1988), 354–362 | DOI | MR | Zbl

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

[6] Pöschel J., Trubowitz E., Inverse spectral theory, Acad. Press, Boston, 1987 | MR | Zbl

[7] Savchuk A. M., Shkalikov A. A., “Operatory Shturma–Liuvillya s potentsialami-raspredeleniyami”, Tr. Mosk. mat. o-va, 64 (2003), 159–219 | MR

[8] Savchuk A. M., Shkalikov A. A., “Inverse problem for Sturm–Liouville operators with distribution potentials: Reconstruction from two spectra”, Russ. J. Math. Phys., 12 (2005), 507–514 | MR | Zbl

[9] Savchuk A. M., Shkalikov A. A., “O sobstvennykh znacheniyakh operatora Shturma–Liuvillya s potentsialami iz prostranstv Soboleva”, Mat. zametki, 80:6 (2006), 864–884 | MR | Zbl

[10] Tartar L., “Interpolation non linéaire et régularité”, J. Funct. Anal., 9 (1972), 469–489 | DOI | MR | Zbl

[11] Yurko V. A., Vvedenie v teoriyu obratnykh spektralnykh zadach, Fizmatlit, M., 2007 | Zbl

[12] Savchuk A. M., Shkalikov A. A., “O svoistvakh otobrazhenii, svyazannykh s obratnymi zadachami Shturma–Liuvillya”, Tr. MIAN, 260, 2008, 227–247 | MR