The Sturm-Liouville inverse spectral problem with boundary conditions depending on the spectral parameter

• Autorzy: Van der Mee C. , Pivovarchik V.
• Języki publikacji: EN

Abstrakty

EN

We present the complete version including proofs of the results announced in [1]. Namely, for the problem of small transversal vibrations of a damped string of nonuniform stiffness with one end fixed we give the description of the spectrum and solve the inverse problem: find the conditions which should be satisfied by a sequence of complex numbers to be the spectrum of a damped string.

Słowa kluczowe

EN

damped vibrations; inhomogeneous strings; quadratic operator pencil; Hermite-Biehler functions

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2005
• Tom: Vol. 25, no. 2
• Strony: 243--260
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Van der Mee C.

• Università di Cagliari, Dipartimento di Matematica e Informatica, Viale Merello 92, 09123 Cagliari, Italy

autor

Pivovarchik V.

• South-Ukrainian Pedagogical State University Staroportofrankovskaya 26, 65091 Odessa, Ukraine.

Bibliografia

• [1] van der Mee C., Pivovarchik V.: A Sturm-Liouville spectral problem with boundary conditions depending on the spectral parameter. Funct. Anal. Appl. 36 (2002), 315-317 [Funkts. Anal. Prilozh. 36 (2002), 74-77 (Russian)].
• [2] Pivovarchik V.: Direct and inverse problem for a damped string. J. Operator Theory 42 (1999), 189-220.
• [3] Pivovarchik V., van der Mee C.: The inverse generalized Regge problem. Inverse Problems 17 (2002), 1831-1845.
• [4] Pivovarchik V.: On positive spectra of one class of polynomial operator pencils. Integral Equations and Operator Theory 19 (1994), 314-326.
• [5] Courant R., Hilbert D.: Methods of Mathematical Physics. Vol. I, New York, Wiley-Interscience 1953.
• [6] Eni V. M.: Stability of the root-number of an analytic operator-function and per­turbations of its characteristic numbers and eigenvectors. Soviet Math. Dokl. 8(2) (1967), 542-545 [Dokl. Akad. Nauk SSSR 173(3) (1967), 1251-1254 (Russian)].
• [7] Gohberg I.C., Sigal E. I.: An operator generalization of the logarithmic residue theorem and the theorem of Rouche. Math. USSR Sbornik 13 (1971), 603-625 [Mat. Sbornik 84 (1971), 607-629 (Russian)].
• [8] Reed M., Simon B.: Methods of Modern Mathematical Physics. Acad. Press, New York - London, 4, 1978.
• [9] Levin B. Ja.: Distribution of Zeros of Entire Functions. Transl. Math. Monogra­phs 5, Providence, RI, Amer. Math. Soc. 1980 [Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956 (Russian)].
• [10] Levin B. Ja., Ostrovskii I.V.: On small perturbations of sets of roots of sinus-type functions. Math. USSR-Izv. 14(1) (1980), 79-101 [Izv. Akad. Nauk SSSR, Ser. Mat., 43 (1979), 87-110, 238 (Russian)].
• [11] Marchenko V. A.: Sturm-Liouville Operators and Applications. Basel and Boston, Birkhauser OT 22, 1986.