The spectrum of Hill’s equation. (English) Zbl 0319.34024


34L99 Ordinary differential operators
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
33E05 Elliptic functions and integrals
Full Text: DOI EuDML


[1] Borg, G.: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Acta Math.78, 1-96 (1945) · Zbl 0063.00523
[2] Buslaev, V., Faddeev, L.: On trace formulas for singular Sturm-Liouville operators. Dokl. Akad. Nauk132, 13-16 (1960) · Zbl 0129.06501
[3] Copson, E.T.: An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press 1960 · Zbl 0188.37901
[4] Dikii, L.A.: Zeta functions for ordinary differential operators on a finite interval. Izv. Akad. Nauk19, 187-200 (1955)
[5] Dikii, L.A.: Trace formulas for Sturm-Liouville differential opertors. AMST18, 81-115 (1958)
[6] Dubrovin, B.A., Novikov, S.P.: Periodic and conditionally periodic analogues of multisoliton solutions of the Korteweg-de Vries equation. Dokl. Akad. Nauk USSR6, 2131-2144 (1974)
[7] Faddeev, L.: The inverse problem of the quantum theory of scattering. Uspehi Mat. Nauk14, 57-119 (1959); J. Math. Phys.4, 72-104 (1963) · Zbl 0091.21902
[8] Faddeev, L., Zaharov, V.: Korteweg-de Vries equation: a completely integrable Hamiltonian system. Funk. Anal. Priloz.5, 18-27 (1971)
[9] Farkas, H., Rauch, H.: Theta Functions with Applications to Riemann Surfaces. Baltimore: Williams and Wilkins 1974 · Zbl 0292.30015
[10] Flaschka, H.: On the inverse problem for Hill’s operator. To appear in Arch. Rat. Mech. Anal. (1975) · Zbl 0376.34016
[11] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Kortewegde Vries equation. Physical review letters19, 1095-1097 (1967) · Zbl 1103.35360
[12] Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Korteweg-de Vries equation and generalizations, 6. Methods for exact solution. Comm. Pure Appl. Math.27, 97-133 (1974) · Zbl 0291.35012
[13] Gelfand, I.M.: On identities for the eigenvalues of a second-order differential operator. Uspehi Mat. Nauk11, 191-198 (1956)
[14] Gelfand, I.M., Levitan, B.M.: On a simple identity for the eigenvalues of a second-order differential operator. Dokl. Akad. Nauk.88, 953-956 (1953)
[15] Gelfand, I.M., Levitan, B.M.: On the determination of a differential equation from its spectral function. Izvest. Akad. Nauk15, 309-360 (1951), AMST1, 253-304 (1955) · Zbl 0044.09301
[16] Goldberg, W.: On the determination of Hill’s equation from its spectrum. To appear in Bull. AMS · Zbl 0331.34026
[17] Goldberg, W.: Necessary and sufficient conditions for determining a Hill’s equation from its spectrum. To appear in Bull. AMS · Zbl 0387.34025
[18] Goldberg, W.: Hill’s equation for a finite number of instability intervals. To appear in J. Math. Anal. Appl.
[19] Göpel, A.: Entwurf einer Theorie der Abel’schen Transcendenten erster Ordnung. Ostwald’s Klassiker der Exacten Wissenschaften. No. 67, Leipzig: Engelmann 1895
[20] Hochstadt, H.: Functiontheoretic properties of the discriminant of Hill’s equation. Math. Zeit.82, 237-242 (1963) · Zbl 0127.04203
[21] Hochstadt, H.: On the determination of Hill’s equation from its spectrum. Arch. Rat. Mech. Anal.19, 353-362 (1965) · Zbl 0128.31201
[22] Its, A.R., Matveev, V.B. Funkt. Anal. i ego Pril.9 (1975)
[23] Jacobi, C.G.L.: Gesammelte Werke. Bd. 7 Supplement: Vorlesungen über Dynamik. Clebsch, A., Reimer, G. (Hrsg.), Berlin: Reimer 1884
[24] Kac, M.: On distributions of certain Wiener functionals. Trans. AMS65, 1-13 (1949) · Zbl 0032.03501
[25] Kac, M.: Can one hear the shape of a drum? Amer. Math. Monthly73, 1-23 (1966) · Zbl 0139.05603
[26] Kac, M., Moerbeke, P. van: On some isospectral second-order differential operators. Proc. Nat. Acad. Sci. USA71, 2350-2351 (1974) · Zbl 0294.34012
[27] Krazer, A., Wirtinger, W.: Abelsche Funktionen und allgemeine Thetafunktionen. Enzyklopädie der Math. Wissenschaften2, art. IIB7. Leipzig: Teubner 1921 · JFM 47.0362.04
[28] Lax, P.: Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl. Math.21, 467-490 (1968) · Zbl 0162.41103
[29] Lax, P.: Periodic solutions of theKdV equations. Lectures in Appl. Math. AMS15, 85-96 (1974)
[30] Lax, P.: Periodic solutions of theKdV equation. Comm. Pure Applied math.28 (1975)
[31] Levinson, N.: The inverse Sturm-Liouville problem. Mat. Tidsskr.4, 25-30 (1949) · Zbl 0045.36402
[32] Magnus, W., Winkler, W.: Hill’s Equation. New York: Interscience-Wiley 1966 · Zbl 0158.09604
[33] Menikoff, A.: The existence of unbounded solutions of the Korteweg-de Vries equation. Comm. Pure Appl. Math.25, 407-432 (1972) · Zbl 0226.35079
[34] Novikov, S.P.: The periodic problem for the Korteweg-de Vries equation. Funkt. Anal. i Ego Pril.8, 54-66 (1974). Translation in Funct. Anal. Jan. 1975, 236-246 · Zbl 0301.54027
[35] Rosenhain, G.: Abhandlung über die Functionen zweier Variabler mit vier Perioden, welche die Inversen sind der ultra-elliptischen Integrale erster Klasse. Ostwald’s Klassiker der Exacten Wissenschaften, no. 65, Leipzig: Engelmann 1895 · JFM 26.0505.02
[36] Siegel, C.L.: Topics in Complex Function Theory. Vol. II. New York: Interscience-Wiley 1971 · Zbl 0211.10501
[37] Strutt, M.J.O.: Lamésche-, Mathiensche-und verwandte Funktionen in Physik und Technik. Berlin: Springer 1932 · Zbl 0005.16005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.