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Inverse spectral problems for differential operators and their applications. (English) Zbl 0952.34001

Analytical Methods and Special Functions. 2. Amsterdam: Gordon and Breach Science Publishers. xiii, 253 p. (2000).
Whereas the inverse problems of recovering the Sturm-Liouville operator from the spectral data on the axis, on the half-axis and from two spectra on a finite interval are investigated comprehensively in the classic papers of I.M. Gel’fand and B. M. Levitan, V. A. Marchenko, M. G. Krein and in tremendous amount of posterior papers (selfadjoint case) and in V. Ljance (nonselfadjoint case) [see V. A. Marchenko, Sturm-Liouville operators and applications. Basel etc.: Birkhäuser Verlag (1986; Zbl 0592.34011); B. M. Levitan, Inverse Sturm-Liouville problems. Utrecht: VNU Science Press (1987; Zbl 0749.34001); K. Chadan and P. C. Sabatier, Inverse problems in quantum scattering theory. New York etc.: Springer-Verlag (1989; Zbl 0681.35088) and M. A. Naimark, Linear differential operators. Part II: Linear differential operators in Hilbert space. New York: Frederick Ungar Publ. Co. (1968; Zbl 0227.34020)] and references therein, the inverse problem for higher-order differential operators has not been investigated so intensively and the present monograph is the first one concerning this topic and therefore important and useful for researchers dealing with inverse problems. The investigation of inverse problems for higher-order differential operators was started by L. A. Sakhnovich, Z. L. Leibenzon, V. I. Matsaev, I. G. Khachatryan. The main part of the book is devoted to the boundary problems generated by the differential expression \[ y^{(n)}+\sum_{k=0}^{n-2}p_k(x)y^{(k)}. \tag{1} \] In Part 1 the following inverse problems are discovered. Using the Weyl matrix (the generalization of the Weyl function of the Sturm-Liouville problem) it is necessary to recover the coefficients \(p_k(x)\). This problem is important because the Weyl function can be constructed from \(n\) spectra of the corresponding boundary problems (in the case of a finite interval) or from the so-called scattering data (in the case of the half-axis). The related problem of existence of a triangular integral operator is discussed. Necessary and sufficient conditions for the unique solvability in the case of a selfadjoint operator (generated by (1)) possessing a simple spectrum are given in explicit form and can be easily checked. In the case of nonselfadjoint operator these conditions are given in implicit form (even in the particular case of Sturm-Liouville operator). The inverse problem for differential operators with singular points is also considered.
In Part 2 the so-called incomplete inverse problem is studied where only some part of the Weyl matrix is given and there is an a priori information about the operator or its spectrum. An uniqueness theorem and an algorithm of solution for a wide class of incomplete inverse problems are presented based on the author’s results. Discrete inverse problems are considered, too.
Some applications are presented in particular to the problem of determination of an elastic beam parameter from given frequencies of its vibrations.
The monograph is equipped with a list of references which should be enlarged with the following references important from our point of view: M. M. Malamud [Trans. Mosc. Math. Soc. 1991, 69-99 (1991); translation from Tr. Mosk. Mat. O.-va 53, 68-97 (1990; Zbl 0745.34030), Trans. Mosc. Math. Soc. 1994, 57-122 (1994); translation from Tr. Mosk. Mat. O.-va 55, 73-148 (1994; Zbl 0846.34088)]; T. Aktosun [SIAM J. Appl. Math. 56, No. 1, 219-231 (1996; Zbl 0844.34016)]; F. Gesztesy and B. Simon [Helv. Phys. Acta 70, No. 1-2, 66-71 (1997; Zbl 0870.34017)]; R. del Rio, F. Gesztesy and B. Simon [Int. Math. Res. Not. 1997, No. 15, 751-758 (1997; Zbl 0898.34075)].

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
34A55 Inverse problems involving ordinary differential equations
47A10 Spectrum, resolvent
34L05 General spectral theory of ordinary differential operators
34B24 Sturm-Liouville theory
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