Guseinov, I. M.; Mammadova, L. I. Reconstruction of the diffusion equation with singular coefficients for two spectra. (English. Russian original) Zbl 1418.34047 Dokl. Math. 90, No. 1, 401-404 (2014); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 457, No. 1, 13-16 (2014). From the text: From the point of view of applications in quantum mechanics [S. Albeverio et al., Solvable models in quantum mechanics. With an appendix by Pavel Exner. 2nd revised ed. Providence, RI: AMS Chelsea Publishing (2005; Zbl 1078.81003)], of interest is the study of direct and inverse problems for the differential equation of diffusion \[ -y''+ [q(x) +\lambda p(x)] y = \lambda^2 y,\qquad 0 \leq x \leq \pi \] where \(q(x)\) is a real function belonging to the space \(L_2[0,\pi]\) \(\lambda\) is a spectral parameter, \(p(x)=\beta \delta (x-a)\), \(\delta\) is the Dirac delta function, \(a\) and \(\beta\) are real numbers, and \(a\in (\pi/2, \pi) \).In this paper, we solve the inverse problem of constructing the function \(q(x)\) from the spectra of the boundary value problems. Cited in 4 Documents MSC: 34B07 Linear boundary value problems for ordinary differential equations with nonlinear dependence on the spectral parameter 34A55 Inverse problems involving ordinary differential equations 81U15 Exactly and quasi-solvable systems arising in quantum theory Keywords:solvable quantum mechanics; spectral inverse problem Citations:Zbl 1078.81003 PDFBibTeX XMLCite \textit{I. M. Guseinov} and \textit{L. I. Mammadova}, Dokl. Math. 90, No. 1, 401--404 (2014; Zbl 1418.34047); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 457, No. 1, 13--16 (2014) Full Text: DOI References: [1] S. Albeverio, F. Gesztezy, R. Hoegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics (Springer, New York, 1988; Mir, Moscow, 1991). · Zbl 0679.46057 [2] Gasymov, M. G.; Guseinov, G. Sh, No article title, Dokl. Akad. Nauk Azerb. SSR, 37, 19-23 (1981) · Zbl 0479.34009 [3] Guseinov, G. Sh, No article title, Dokl. Akad. Nauk SSSR, 285, 1292-1296 (1985) [4] Guseinov, I. M.; Nabiev, I. M., No article title, Sb.: Math., 198, 1579-1598 (2007) · Zbl 1152.34005 [5] Nabiev, I. M., No article title, Dokl. Math., 76, 527-529 (2007) · Zbl 1159.34307 [6] Sadovnichii, V. A.; Sultanaev, Ya T.; Akhtyamov, A. M., No article title, Dokl. Math., 79, 169-171 (2009) · Zbl 1347.34046 [7] Hryniv, R.; Pronska, N., No article title, Inverse Problems, 28, 085008 (2012) · Zbl 1256.34011 [8] Amirov, R. Kh; Nabiev, A. A., No article title, Abstr. Appl. Anal., 2013, 361989 (2013) · Zbl 1279.34016 [9] Yurko, V., No article title, Integral Transforms Spec. Funct., 10, 141-164 (2000) · Zbl 0989.34015 [10] Amirov, R. Kh; Yurko, V. A., No article title, Ukr. Mat. Zh., 53, 1443-1457 (2001) · Zbl 1015.34072 [11] Freiling, G.; Yurko, V., No article title, Inverse Problems, 18, 757-773 (2002) · Zbl 1012.34083 [12] Guseinov, I. M.; Pashaev, R. T., No article title, Uspekhi Mat. Nauk, 57, 147-148 (2002) [13] Jamshidipour, A. H.; Huseynov, H. M., No article title, Trans. Nat. Acad. Sci. Azerb., 22, 89-100 (2012) [14] Huseynov, H. M.; Osmanli, J. A., No article title, J. Math. Phys. Anal. Geom., 9, 332-359 (2013) · Zbl 1298.34167 [15] V. A. Marchenko, Sturm-Liouville Operators and Their Applications (Naukova Dumka, Kiev, 1977) [in Russian]. · Zbl 0399.34022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.