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

The author considers the following Sturm-Liouville-like problem:

y '' +(λ 2 -q-iλp)y=0on(0,a),y(0)=y(a)=0·

Here, λ is the spectral parameter, qL 2 ((0,a),) such that the operator A defined by Ay=-y '' +qy on {yW 2,2 ((0,a),); 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(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 '' +(λ 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:
34B24Sturm-Liouville theory
34A55Inverse problems of ODE
34L40Particular ordinary differential operators
34K29Inverse problems in theory of functional-differential equations
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.