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Dressing chains and the spectral theory of the Schrödinger operator. (English. Russian original) Zbl 0813.35099
Funct. Anal. Appl. 27, No. 2, 81-96 (1993); translation from Funkts. Anal. Prilozh. 27, No. 2, 1-21 (1993).
The authors consider the sequence of Schrödinger operators $L_ i = - (D + f_ i) (D-f_ i) = - \Delta + u_ i, \quad u_ i = f_ i' + f^ 2_ i, \tag{1}$ where the functions $$f_ i$$ satisfy the following system of ordinary differential equations, $(f_ i + f_{i+1})' = f_ i^ 2 - f_{i+1}^ 2 + \alpha_ i, \quad f_{i+N} = f_ i, \quad \alpha_{i+N} = \alpha_ i, \tag{2}$ $$1 \leq i \leq N$$, called a dressing chain. Here $$\alpha_ 1, \dots, \alpha_ N$$ are constant parameters. The authors study the system (2) and prove, for example, that if $$N$$ is odd and $$\alpha_ 1 + \cdots + \alpha_ N = 0$$ then the chain (2) is a completely integrable Hamiltonian system. The spectral properties of the corresponding Schrödinger operators (1) are investigated as well.

##### MSC:
 35Q40 PDEs in connection with quantum mechanics 35P05 General topics in linear spectral theory for PDEs 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35J10 Schrödinger operator, Schrödinger equation
##### Keywords:
Schrödinger operators; dressing chain; spectrum
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##### References:
 [1] S. P. Novikov, ?Periodic problem for the Korteweg?de Vries equation. I,? Funkts. Anal. Prilozhen.,8, No. 3, 54-66 (1974). · Zbl 0301.54027 [2] 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). · Zbl 0326.35011 [3] P. D. Lax, ?Periodic solutions of Korteweg?de Vries equation,? Comm. Pure Appl. Math.,28, 141-188 (1975). · Zbl 0302.35008 [4] H. P. McKean and P. van Moerbeke, ?The spectrum of Hill’s equation,? Invent. Math.,30, 217-274 (1975). · Zbl 0319.34024 [5] V. A. Marchenko, Sturm?Liouville Operators and their Applications [in Russian], Naukova Dumka, Kiev (1977). · Zbl 0399.34022 [6] G. Darboux, ?Sur la representations spherique des surfaces,? Compt. Rend.,94, 1343-1345 (1882). [7] M. M. Crum, ?Associated Sturm?Liouville systems,? Quart. J. Math. Ser. 2,6, 121-127 (1955). · Zbl 0065.31901 [8] A. B. Shabat, ?One-dimensional perturbations of a differential operator and inverse scattering problem,? Selecta Math. Soviet.,4, No. 1, 19-35 (1985). · Zbl 0569.34026 [9] P. Deift, ?Application of a commutation formula,? Duke Math. J.,45, 267-310 (1978). · Zbl 0392.47013 [10] M. Adler and J. Moser, ?On a class of polynomials connected with the Korteweg?de Vries equation,? Comm. Math. Phys.,61, 1-30 (1978). · Zbl 0428.35067 [11] A. B. Shabat, ?The infinite-dimensional dressing dynamical system,? Inverse Problems,6, 303-308 (1992). · Zbl 0762.35098 [12] A. B. Shabat and R. I. Yamilov, ?Symmetries of nonlinear chains,? Algebra Analiz,2, No. 2, 183-208 (1990). [13] 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). · Zbl 0423.70018 [14] J. L. Burchnall and T. W. Chaundy, ?Commutative ordinary differential operators,? Proc. London Soc. Ser. 2,21, 420-440 (1923). · JFM 49.0311.03 [15] E. L. Ince, Ordinary Differential Equations, Dover, New York (1947). · Zbl 0063.02971 [16] F. Magri, ?A simple model of an integrable Hamiltonian equation,? J. Math. Phys.,19, 1156-1162 (1978). · Zbl 0383.35065 [17] I. M. Gelfand and I. Ya. Dorfman, ?Hamiltonian operators and connected algebraic structures,? Funkts. Anal. Prilozhen.,13, No. 4, 13-30 (1979). [18] M. Antonowicz, A. P. Fordy, and S. Wojciechowski, ?Integrable stationary flows: Miura maps and bi-Hamiltonian structures,? Phys. Lett. A,124, 143-150 (1987). [19] 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). · Zbl 0632.35063 [20] P. Santini, ?Solvable nonlinear algebraic equations,? Inverse Problems,6 (1990). · Zbl 0737.35107 [21] 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). [22] M. Adler and P. van Moerbeke, Algebraic Integrable Systems: a Systematic Approach. Perspectives in Math., Academic Press, Boston (1989). [23] V. E. Adler, ?Recuttings of polygons,? Funkts. Anal. Prilozhen.,27, No. 2, 79-82 (1993). [24] F. J. Bureau, ?Integration of some nonlinear systems of ordinary differential equations,? Annali Mat. Pura Appl. (IV),44, 345-360 (1972). · Zbl 0293.34007 [25] 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). · Zbl 0516.35071 [26] H. Bateman and A. Erdély, Higher Transcendental Functions. Vol. 3, McGraw-Hill (1955). [27] S. P. Novikov (ed.), Theory of Solitons [in Russian], Nauka, Moscow (1980). · Zbl 0598.35003 [28] H. P. McKean and E. Trubowitz, ?The spectral class of the quantum mechanical harmonic oscillator,? Comm. Math. Phys.,82, 471-495 (1982). · Zbl 0493.34012 [29] 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). · Zbl 0625.34021
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