Reconstruction techniques for classical inverse Sturm-Liouville problems

Rundell, William; Sacks, Paul E.

doi:10.1090/S0025-5718-1992-1106979-0

Reconstruction techniques for classical inverse Sturm-Liouville problems
HTML articles powered by AMS MathViewer

by William Rundell and Paul E. Sacks PDF

Math. Comp. 58 (1992), 161-183 Request permission

Abstract:

This paper gives constructive algorithms for the classical inverse Sturm-Liouville problem. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. Two methods of solving this latter problem are then provided, and numerical examples are presented.

References

L.-E. Andersson, Algorithms for solving inverse eigenvalue problems for Sturm-Liouville equations, Inverse methods in action (Montpellier, 1989) Inverse Probl. Theoret. Imaging, Springer, Berlin, 1990, pp. 138–145. MR 1107960
Victor Barcilon, Iterative solution of the inverse Sturm-Liouville problem, J. Mathematical Phys. 15 (1974), 429–436. MR 342069, DOI 10.1063/1.1666664
Paul B. Bailey, Burton S. Garbow, Hans G. Kaper, and Anton Zettl, Eigenvalue and eigenfunction computations for Sturm-Liouville problems, ACM Trans. Math. Software 17 (1991), no. 4, 491–499. MR 1140036, DOI 10.1145/210232.210238
Göran Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946), 1–96 (German). MR 15185, DOI 10.1007/BF02421600
I. M. Gel′fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Amer. Math. Soc. Transl. (2) 1 (1955), 253–304. MR 0073805
Ole H. Hald, The inverse Sturm-Liouville problem with symmetric potentials, Acta Math. 141 (1978), no. 3-4, 263–291. MR 505878, DOI 10.1007/BF02545749
Ole H. Hald, The inverse Sturm-Liouville problem and the Rayleigh-Ritz method, Math. Comp. 32 (1978), no. 143, 687–705. MR 501963, DOI 10.1090/S0025-5718-1978-0501963-2
Ole H. Hald and Joyce R. McLaughlin, Solutions of inverse nodal problems, Inverse Problems 5 (1989), no. 3, 307–347. MR 999065
Harry Hochstadt, The inverse Sturm-Liouville problem, Comm. Pure Appl. Math. 26 (1973), 715–729. Collection of articles dedicated to Wilhelm Magnus. MR 330607, DOI 10.1002/cpa.3160260514
Harry Hochstadt and Burton Lieberman, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math. 34 (1978), no. 4, 676–680. MR 470319, DOI 10.1137/0134054
E. L. Isaacson and E. Trubowitz, The inverse Sturm-Liouville problem. I, Comm. Pure Appl. Math. 36 (1983), no. 6, 767–783. MR 720593, DOI 10.1002/cpa.3160360604
E. L. Isaacson, H. P. McKean, and E. Trubowitz, The inverse Sturm-Liouville problem. II, Comm. Pure Appl. Math. 37 (1984), no. 1, 1–11. MR 728263, DOI 10.1002/cpa.3160370102
Norman Levinson, The inverse Sturm-Liouville problem, Mat. Tidsskr. B 1949 (1949), 25–30. MR 32067
Norman Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, Vol. 26, American Mathematical Society, New York, 1940. MR 0003208
B. M. Levitan, Inverse Sturm-Liouville problems, VSP, Zeist, 1987. Translated from the Russian by O. Efimov. MR 933088
Joyce R. McLaughlin and George H. Handelman, Sturm-Liouville inverse eigenvalue problems, Mechanics today, Vol. 5, Pergamon, Oxford-New York, 1980, pp. 281–295. MR 591510
Joyce R. McLaughlin, Analytical methods for recovering coefficients in differential equations from spectral data, SIAM Rev. 28 (1986), no. 1, 53–72. MR 828436, DOI 10.1137/1028003
Joyce R. McLaughlin, Inverse spectral theory using nodal points as data—a uniqueness result, J. Differential Equations 73 (1988), no. 2, 354–362. MR 943946, DOI 10.1016/0022-0396(88)90111-8
Joyce R. McLaughlin, Stability theorems for two inverse spectral problems, Inverse Problems 4 (1988), no. 2, 529–540. MR 954908
Jürgen Pöschel and Eugene Trubowitz, Inverse spectral theory, Pure and Applied Mathematics, vol. 130, Academic Press, Inc., Boston, MA, 1987. MR 894477
Paul E. Sacks, An iterative method for the inverse Dirichlet problem, Inverse Problems 4 (1988), no. 4, 1055–1069. MR 966772
Takashi Suzuki, Gel′fand-Levitan’s theory, deformation formulas and inverse problems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), no. 2, 223–271. MR 802892