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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.

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

Citations:

Zbl 1078.81003
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Full Text: DOI

References:

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