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Equivalence of inverse Sturm--Liouville problems with boundary conditions rationally dependent on the eigenparameter. (English) Zbl 1046.34021
Inverse spectral problems are considered for the boundary value problem $$ -y''+q(x)y=\lambda y,\quad 0<x<1, $$ $$ y(0)\cos\alpha=y'(0)\sin\alpha,\quad y'(1)=f(\lambda)y(1). $$ Here, $q\in AC[0,1]$, $\alpha\in[0,\pi)$, $f(\lambda)=h(\lambda)/g(\lambda)$, where $g$ and $h$ are polynomials with real coefficients and no common zeros. Uniqueness theorems are proved for two inverse problems of recovering the potential $q$ and the coefficients of the boundary conditions from the Weyl function and from two spectra.

34A55Inverse problems of ODE
34B24Sturm-Liouville theory
34B07Linear boundary value problems with nonlinear dependence
Full Text: DOI
[1] Ashbaugh, M. S.; Benguria, R. D.: Eigenvalue ratios for Sturm--Liouville operators. J. differential equations 103, 205-219 (1993) · Zbl 0785.34027
[2] Atkinson, F. V.: Discrete and continuous boundary value problems. (1964) · Zbl 0117.05806
[3] Atkinson, F. V.; Fulton, C.: Asymptotic formulae for eigenvalues of limit circle problems on a half line. Ann. mat. Pura appl. (4) 135, 363-398 (1983) · Zbl 0597.34024
[4] Bennewitz, C.: A proof of the local borg--marchenko theorem. Comm. math. Phys. 218, 131-132 (2001) · Zbl 0982.34021
[5] Binding, P. A.; Browne, P. J.; Seddighi, K.: Sturm--Liouville problems with eigenparameter dependent boundary conditions. Proc. Edinburgh math. Soc. (2) 37, 57-72 (1993) · Zbl 0791.34023
[6] Binding, P. A.; Browne, P. J.; Watson, B. A.: Inverse spectral problems for Sturm--Liouville equations with eigenparameter dependent boundary conditions. J. London math. Soc. 62, 161-182 (2000) · Zbl 0960.34010
[7] P.A. Binding, P.J. Browne, B.A. Watson, Spectral isomorphisms between generalized Sturm--Liouville problems, Oper. Theory Adv. Appl., in press · Zbl 1038.34027
[8] Borg, G.: Eine umkehrung der Sturm--liouvilleschen eigenwertaufgabe. Bestimmung der differentialgleichung durch die eigenwerte. Acta math. 78, 1-96 (1946) · Zbl 0063.00523
[9] Dijksma, A.: Eigenfunction expansions for a class of J-selfadjoint ordinary differential operators with boundary conditions containing the eigenvalue parameter. Proc. roy. Soc. Edinburgh sect. A 87, 1-27 (1980) · Zbl 0434.47037
[10] Eastham, M. S. P.: The spectral theory of periodic differential equations. (1973) · Zbl 0287.34016
[11] Freiling, G.; Yurko, V.: Inverse Sturm--Liouville problems and their applications. (2001) · Zbl 1037.34005
[12] Gesztesy, F.; Simon, B.; Teschl, G.: Zeros of the Wronskian and renormalized oscillation theory. Amer. J. Math. 118, 571-594 (1996) · Zbl 0858.47027
[13] Hochstadt, H.: On inverse problems associated with second-order differential operators. Acta math. 119, 173-192 (1967) · Zbl 0155.13002
[14] Kac, I. S.; Krein, M. G.: On the spectral functions of the string. Amer. math. Soc. transl. 103, 19-102 (1974) · Zbl 0291.34017
[15] Levitan, B. M.; Gasymov, M. G.: Determination of a differential equation from two of its spectra. Russian math. Surveys 19, 1-63 (1964) · Zbl 0145.10903
[16] Marc\breve{}enko, V. A.: Some questions in the theory of one-dimensional linear differential operators of the second order, part I. Amer. math. Soc. transl. Ser. 2 101, 1-104 (1973)
[17] Naimark, M. A.: Linear differential operators. (1967) · Zbl 0219.34001
[18] Poisson, S. D.: Memoire sur la maniere d’exprimer LES functions par des series periodiques. J. ecole poly-technique 18, 417-489 (1820)
[19] Prüfer, H.: Neue herleitung der Sturm liouvilleschen reihenentwicklung stetiger funktionen. Math. ann. 95, 409-518 (1926) · Zbl 52.0455.01
[20] Rundell, W.; Sacks, P. E.: Reconstruction techniques for classical inverse Sturm--Liouville problems. Math. comp. 58, 161-183 (1992) · Zbl 0745.34015
[21] Russakovskii, E. M.: Operator treatment of boundary problems with spectral parameters entering via polynomials in the boundary conditions. Funct. anal. Appl. 9, 358-359 (1975)
[22] Simon, B.: A new approach to inverse spectral theory, I. Fundamental formalism. Ann. of math. 150, 1029-1057 (1999) · Zbl 0945.34013
[23] Titchmarsh, E. C.: The theory of functions. (1939) · Zbl 0022.14602
[24] Titchmarsh, E. C.: On expansions in eigenfunctions (III). Quart. J. Math. Oxford ser. (2) 12, 33-50 (1941) · Zbl 67.0327.02
[25] Weyl, H.: Über gewohnliche differentialgleichungen mit singularitäten und die zugehörigen entwicklungen willkürlicher funktionen. Math. ann. 68, 220-269 (1910) · Zbl 41.0343.01