Geodesic
The Dynamics of Zeros of Finite-Gap Solutions of the Schrödinger Equation
Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 4, pp. 8-19 
A. A. Akhmetshin; Yu. S. Vol'vovskii. The Dynamics of Zeros of Finite-Gap Solutions of the Schrödinger Equation. Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 4, pp. 8-19. http://geodesic.mathdoc.fr/item/FAA_2001_35_4_a1/
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     title = {The {Dynamics} of {Zeros} of {Finite-Gap} {Solutions} of the {Schr\"odinger} {Equation}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

We study a system of particles on a Riemann surface with a puncture. This system describes the behavior of zeros of finite-gap solutions of the Schrödinger equation corresponding to a degenerate hyperelliptic curve. We show that this system is Hamiltonian and integrable by constructing action-angle type coordinates.

[1] Van Diejen J. F., Puschmann H., “Reflectionless Schrödinger operators, the dynamics of zeros, and the solitonic Sato formula”, Duke Math. J., 104:2 (2000), 269–318 | DOI | MR | Zbl

[2] Ruijsenaars S. N. M., Schneider H., “A new class of integrable systems and its relation to solitons”, Ann. Physics, 170:2 (1986), 370–405 | DOI | MR | Zbl

[3] Dubrovin B. A., Matveev V. B., Novikov S. P., “Nelineinye uravneniya tipa Kortevega–de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, UMN, 31:1(187) (1976), 55–136 | MR | Zbl

[4] Dubrovin B. A., “Periodicheskie zadachi dlya uravneniya Kortevega–de Friza v klasse konechnozonnykh potentsialov”, Funkts. analiz i ego pril., 9:3 (1975), 41–52 | MR

[5] Krichever I. M., “Potentsialy s nulevym koeffitsientom otrazheniya na fone konechno-zonnykh potentsialov”, Funkts. analiz i ego pril., 9:2 (1975), 77–79 | MR

[6] Krichever I. M., “Spektralnaya teoriya “konechnozonnykh” nestatsionarnykh operatorov Shrëdingera. Nestatsionarnaya model Paierlsa”, Funkts. analiz i ego pril., 20:3 (1986), 42–54 | MR | Zbl

[7] Bateman H., Erdelyi A., Higher Transcendental Functions, Vol. II, McGraw-Hill, 1953 | MR

[8] Veselov A. P., Novikov S. P., “O skobkakh Puassona, sovmestimykh s algebraicheskoi geometriei i dinamikoi Kortevega–de Friza na mnozhestve konechnozonnykh potentsialov”, DAN SSSR, 266:3 (1982), 533–537 | MR

[9] Krichever I. M., Phong D., “Symplectic forms in the theory of solitons”, Surveys in differential geometry: integral systems, Surv. Differ. Geom. IV, 1998, 239–313 | DOI | MR | Zbl