Finite dimensional four spectra inverse problem

O. M. Martynyuk; V. N. Pivovarchik

We find necessary and suffcient conditions for four finite sequences of positive numbers to be certain parts of spectra of the Dirichlet-Dirichlet, Dirichlet-Neumann, Neumann-Dirichlet and Neumann-Neumann boundary value problems generated by the same Stieltjes string recurrence relations.

1. V.A. Ambarzumian, Uber eine Frage der Eigenwerttheorie, Zeitschrift fur Physik, 53 (1929), 690–695.

2. G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78 (1946), 1–96.

3. M. Horvath, On the inverse spectral theory of SrЁodinger and Dirac operators, Trans. Amer. Math. Soc., 353 (2001), №10, 4155–4171.

4. M. Horvath, Inverse spectral problems and closed exponential systems, Ann. of Math., 162 (2005), 885– 918.

5. F. Gesztesy, R. del Rio, B. Simon, Inverse spectral analysis with partial information on the potential. III. Updating boundary conditions, Int. Math. Res. Not. IMRN, 15 (1997), 751–758.

6. R.F. Gantmakher, M.G. Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, GITTL, Moscow–Leningrad, 1950. (in Russian); German transl.: Akademie Verlag, Berlin, 1960.

7. C.-K. Law, V. Pivovarchik, W.C. Wang, Apolynomial identity and its application to inverse spectral problems in Stieltjes strings, to appear in Operators and Matrices.

8. V. Pivovarchik, An inverse problem by eigenvalues of four spectra, J. Math. Anal. Appl., 396 (2012), №2, 715–723.

	Finite dimensional four spectra inverse problem
Author	O. M. Martynyuk, V. N. Pivovarchik v.pivovarchik@paco.net South Ukrainian National Pedagogical University
Abstract	We find necessary and suffcient conditions for four finite sequences of positive numbers to be certain parts of spectra of the Dirichlet-Dirichlet, Dirichlet-Neumann, Neumann-Dirichlet and Neumann-Neumann boundary value problems generated by the same Stieltjes string recurrence relations.
Keywords	Lagrange identity; Stieltjes string; eigenvalues; continued fraction
Reference	1. V.A. Ambarzumian, Uber eine Frage der Eigenwerttheorie, Zeitschrift fur Physik, 53 (1929), 690–695. 2. G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe, Acta Math., 78 (1946), 1–96. 3. M. Horvath, On the inverse spectral theory of SrЁodinger and Dirac operators, Trans. Amer. Math. Soc., 353 (2001), №10, 4155–4171. 4. M. Horvath, Inverse spectral problems and closed exponential systems, Ann. of Math., 162 (2005), 885– 918. 5. F. Gesztesy, R. del Rio, B. Simon, Inverse spectral analysis with partial information on the potential. III. Updating boundary conditions, Int. Math. Res. Not. IMRN, 15 (1997), 751–758. 6. R.F. Gantmakher, M.G. Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, GITTL, Moscow–Leningrad, 1950. (in Russian); German transl.: Akademie Verlag, Berlin, 1960. 7. C.-K. Law, V. Pivovarchik, W.C. Wang, Apolynomial identity and its application to inverse spectral problems in Stieltjes strings, to appear in Operators and Matrices. 8. V. Pivovarchik, An inverse problem by eigenvalues of four spectra, J. Math. Anal. Appl., 396 (2012), №2, 715–723.
Pages	73-80
Volume	41
Issue	1
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html