×

Reconstruction of the potential of the Sturm-Liouville equation from three spectra of boundary value problems. (English. Russian original) Zbl 0948.34015

Funct. Anal. Appl. 33, No. 3, 233-235 (1999); translation from Funkts. Anal. Prilozh. 33, No. 3, 87-90 (1999).
The author considers the following Sturm-Liouville-like problem: \[ y''+ (\lambda^2- q-i\lambda p)y= 0\quad\text{on }(0,a),\quad y(0)= y(a)= 0. \] Here, \(\lambda\) is the spectral parameter, \(q\in L^2((0,a), \mathbb{R})\) such that the operator \(A\) defined by \(Ay= -y''+ qy\) on \(\{y\in W^{2,2}((0, a),\mathbb{R})\); \(y(0)= y(a)= 0\}\) is strictly positive, and \(p\) is a positive constant \(c\) on \((0,b)\) and \(0\) on \((b,a)\) for some \(b\in (0,a)\).
Two associated problems are introduced: one consists of the above differential equation restricted to \((0,b)\) and the boundary condition \(y(0)= y(b)= 0\), while the other of the equation \(y''+ (\lambda^2- q)y= 0\) on \((b,a)\) and the condition \(y(b)= y(a)= 0\).
The author gives a procedure for reconstructing, from the spectra of these three problems, the constants \(a\), \(b\), \(c\) and the function \(q\) and hence proves the following result: the constants \(a\), \(b\), \(c\) and the function \(q\) in the original problem are uniquely determined by the corresponding three spectra.

MSC:

34B24 Sturm-Liouville theory
34A55 Inverse problems involving ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34K29 Inverse problems for functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Jaulent and C. Jean, Commun. Math. Phys.,28, 177–220 (1972). · doi:10.1007/BF01645775
[2] M. Jaulent, J. Math. Phys.,17, No. 7, 1351–1360 (1976). · doi:10.1063/1.523064
[3] M. Jaulent and C. Jean, Ann. Inst. H. Poincaré, sec. A,25, No. 2, 105–137 (1976).
[4] T. Aktosun, M. Klaus, and C. van der Mee, J. Math. Phys.,39, No. 4, 1957–1992 (1998). · Zbl 1001.34074 · doi:10.1063/1.532271
[5] M. G. Gasymov, and G. Sh. Guseinov, Dokl. AN Azerb. SSR,37, No. 2, 19–23 (1981).
[6] V. N. Pivovarchik, Integral Equations and Operator Theory,34, 234–243 (1999). · Zbl 0948.34014 · doi:10.1007/BF01236474
[7] F. Gesztesy and B. Simon, To be published in: Birman Birthday Volume in Advances in Mathematical Sciences (V. Buslaev and M. Solomyak, eds.), Amer. Math. Soc., Providence, RI.
[8] V. A. Marchenko, Sturm-Liouville Operators and Applications, in: Operator Theory: Advances and Applications, Vol. 22, Birkhäuser, Basel-Boston, 1986.
[9] B. Ya. Levin, Lectures on Entire Functions, Translations of Mathematical Monographs, Vol. 150, 1996.
[10] B. M. Levitan and M. G. Gasymov, Usp. Mat. Nauk,19, No. 2, (116), 3–63 (1964).
[11] B. M. Levitan, Inverse Sturm-Liouville Problems [in Russian], Nauka, Moscow, 1984. · Zbl 0575.34001
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.