Dressing chains and the spectral theory of the Schr�dinger operator

А. П. Веселов, A. B. Shabat

Tóm tắt

Từ khóa


Tài liệu tham khảo

S. P. Novikov, ?Periodic problem for the Korteweg?de Vries equation. I,? Funkts. Anal. Prilozhen.,8, No. 3, 54?66 (1974).

B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, ?Nonlinear equations of KdV type, finite-zone linear operators and abelian varieties,? Usp. Mat. Nauk,31, No. 1, 55?136 (1976).

P. D. Lax, ?Periodic solutions of Korteweg?de Vries equation,? Comm. Pure Appl. Math.,28, 141?188 (1975).

H. P. McKean and P. van Moerbeke, ?The spectrum of Hill's equation,? Invent. Math.,30, 217?274 (1975).

V. A. Marchenko, Sturm?Liouville Operators and their Applications [in Russian], Naukova Dumka, Kiev (1977).

G. Darboux, ?Sur la representations spherique des surfaces,? Compt. Rend.,94, 1343?1345 (1882).

M. M. Crum, ?Associated Sturm?Liouville systems,? Quart. J. Math. Ser. 2,6, 121?127 (1955).

A. B. Shabat, ?One-dimensional perturbations of a differential operator and inverse scattering problem,? Selecta Math. Soviet.,4, No. 1, 19?35 (1985).

P. Deift, ?Application of a commutation formula,? Duke Math. J.,45, 267?310 (1978).

M. Adler and J. Moser, ?On a class of polynomials connected with the Korteweg?de Vries equation,? Comm. Math. Phys.,61, 1?30 (1978).

A. B. Shabat, ?The infinite-dimensional dressing dynamical system,? Inverse Problems,6, 303?308 (1992).

A. B. Shabat and R. I. Yamilov, ?Symmetries of nonlinear chains,? Algebra Analiz,2, No. 2, 183?208 (1990).

A. P. Veselov, ?On the Hamiltonian formalism for the Novikov?Krichever equation of commutativity of two operators,? Funkts. Anal. Prilozhen.,13, No. 1, 1?7 (1979).

J. L. Burchnall and T. W. Chaundy, ?Commutative ordinary differential operators,? Proc. London Soc. Ser. 2,21, 420?440 (1923).

E. L. Ince, Ordinary Differential Equations, Dover, New York (1947).

F. Magri, ?A simple model of an integrable Hamiltonian equation,? J. Math. Phys.,19, 1156?1162 (1978).

I. M. Gelfand and I. Ya. Dorfman, ?Hamiltonian operators and connected algebraic structures,? Funkts. Anal. Prilozhen.,13, No. 4, 13?30 (1979).

M. Antonowicz, A. P. Fordy, and S. Wojciechowski, ?Integrable stationary flows: Miura maps and bi-Hamiltonian structures,? Phys. Lett. A,124, 143?150 (1987).

J. Weiss, ?Periodic fixed points of Bäcklund transformations and the KdV equation,? J. Math. Phys.,27 (11, 2647?2656 (1986);28 (9), 2025?2039 (1987).

P. Santini, ?Solvable nonlinear algebraic equations,? Inverse Problems,6 (1990).

A. P. Veselov, ?On the growth of the number of images of the point under the iterations of the multivalued mapping,? Mat. Zametki,49, No. 2, 29?35 (1991).

M. Adler and P. van Moerbeke, Algebraic Integrable Systems: a Systematic Approach. Perspectives in Math., Academic Press, Boston (1989).

V. E. Adler, ?Recuttings of polygons,? Funkts. Anal. Prilozhen.,27, No. 2, 79?82 (1993).

F. J. Bureau, ?Integration of some nonlinear systems of ordinary differential equations,? Annali Mat. Pura Appl. (IV),44, 345?360 (1972).

F. Ehlers and H. Knoerrer, ?An algebro-geometric interpretation of the Bäcklund transformation for the Korteweg?de Vries equation,? Comm. Math. Helv.,57, No. 1, 1?10 (1982).

H. Bateman and A. Erdély, Higher Transcendental Functions. Vol. 3, McGraw-Hill (1955).

S. P. Novikov (ed.), Theory of Solitons [in Russian], Nauka, Moscow (1980).

H. P. McKean and E. Trubowitz, ?The spectral class of the quantum mechanical harmonic oscillator,? Comm. Math. Phys.,82, 471?495 (1982).

B. M. Levitan, ?On the Sturm?Liouville operators on the whole line with one and the same spectrum,? Mat. Sb.,132, No. 1, 73?103 (1987).