On Hill's equation with a singular complex-valued potential

Proceedings of the London Mathematical Society

Volumn 76, Issue 3, 1998, Pages 603-633

  Weikard, Rudi ^a  

^a UNIVERSITY OF ALABAMA AT BIRMINGHAM   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0040546979     PISSN: 00246115     EISSN: None     Source Type: Journal    
DOI: 10.1112/S0024611598000343     Document Type: Article

References (28)

1. M. ABRAMOWTZ and I. A. STEGUN, Handbook of mathematical functions (Dover, New York, 1972).
2. S. I. AL'BER, 'Investigation of equations of Korteweg-de Vries type by the method of recurrence relations', J. London Math. Soc. (2) 19 (1979) 467-480.
3. E. D. BELOKOLOS, A. I. BOBENKO, V. Z. ENOL'SKII, A. R. ITS, and V. B. MATVEEV, Algebro-geometric approach to nonlinear integrable equations (Springer, Berlin, 1994).
4. G. D. BIRKHOFF, 'Existence and oscillation theorem for a certain boundary value problem', Trans. Amer. Math. Soc. 10 (1909) 259-270.
5. B. BIRNIR, 'Complex Hill's equation and the complex periodic Korteweg-de Vries equations', Comm. Pure Appl. Math. 39 (1986) 1-49.
6. J. L. BURCHNALL and T. W. CHAUNDY, 'Commutative ordinary differential operators', Proc. London Math. Soc. (2) 21 (1923) 420-440.
7. J. L. BURCHNALL and T. W. CHAUNDY, 'Commutative ordinary differential operators', Proc. Roy. Soc. London A 118 (1928) 557-583.
8. L. A. DICKEY, Soliton equations and Hamiltonian systems (World Scientific, Singapore, 1991).
9. B. A. DUBROVIN, 'Periodic problems for the Korteweg-de Vries equation in the class of finite band potentials', Funct. Anal. Appl. 9 (1975) 215-223.
10. M. S. P. EASTHAM, The spectral theory of periodic differential equations (Scottish Academic Press, Edinburgh, 1973).
11. I. M. GEL'FAND and L. A. DIKII, 'Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations', Russian Math. Surveys 30:5 (1975) 77-113.
12. F. GESZTESY and R. WEIKARD, 'Lamé potentials and the stationary (m)KdV hierarchy', Math. Nachr. 176 (1995) 73-91.
13. F. GESZTESY and R. WEIKARD, 'Treibich-Verdier potentials and the stationary (m)KdV hierarchy', Math. Z. 219 (1995) 451-476.
14. F. GESZTESY and R. WEIKARD, 'On Picard potentials', Differential Integral Equations 8 (1995) 1453-1476.
15. F. GESZTESY and R. WEIKARD, 'Picard potentials and Hill's equation on a torus', Acta Math. 176 (1996) 73-107.
16. G. HAMEL, 'Über die lineare Differentialgleichung zweiter Ordnung mit periodischen Koeffizienten', Math. Ann. 73 (1913) 371-412.
17. O. HAUPT, 'Über lineare homogene Differentialgleichungen 2. Ordnung mit periodischen Koeffizienten', Math. Ann. 79 (1919) 278-285.
18. E. L. INCE, 'Further investigations into the periodic Lamé functions', Proc. Roy. Soc. Edinburgh 60 (1940) 83-99.
19. A. R. ITS and V. B. MATVEEV, 'Schrödinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation', Theoret. Math. Phys. 23 (1975) 343-355.
20. P. LAX, 'Integrals of nonlinear equations of evolution and solitary waves', Comm. Pure Appl. Math. 21 (1968) 467-490.
21. P. LAX, 'Periodic solutions of the KdV equation', Comm. Pure Appl. Math. 28 (1975) 467-490.
22. A. LIAPOUNOFF, 'Sur une équation transcendante et les équations différentielles linéaires du second ordre à coefficients périodiques', Comptes Rendus 128 (1899) 1085-1088.
23. H. P. MCKEAN and P. VAN MOERBEKE, 'The spectrum of Hill's equation', Invent. Math. 30 (1975) 217-274.
24. M. A. NAIMARK, Linear differential operators (Frederick Ungar, New York, 1967).
25. S. P. NOVIKOV, 'The periodic problem for the Korteweg-de Vries equation', Funct. Anal. Appl. 8 (1974) 236-246.
26. F. S. ROFE-BEKETOV, 'The spectrum of non-selfadjoint differential operators with periodic coefficients', Soviet Math. Dokl. 4 (1963) 1563-1566.
27. R. WEIKARD, 'On second order linear differential equations with inverse square singularities', Differential and integral operators, proceedings of IWOTA 95 (eds R. Mennicken and C. Treuer, Birkhäuser, Basle, 1997), pp. 315-324.
28. K. YOSIDA, Lectures on differential and integral equations (Dover, New York, 1991).