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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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