Geodesic
Schr\"odinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg--de~Vries equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 23 (1975) no. 1, pp. 51-68

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Explicit description of periodic potentials for which the corresponding Schrodinger operator $N$ possesses only the finite number of energy gaps is obtained. Using this result the solution of the Korteveg–de Vries equation with the “finite-gap” initial condition is expressed, by means of the $N$-dimensional $\Theta$-function, $N$ being the number of the nondegenerate energy gaps. The following characteristic property of the $N$-gap periodic potentials and the $N$-soliton decreasing potentials is discovered: the existence of two solutions $\psi_1(x,\lambda), \psi_2(x,\lambda)$ of the Schrodinger equation, for which the product $\psi_1,\psi_2$ is the polynomial $P$ ($\operatorname{deg}P=N$. $N$ is the number of gaps or the number of bound states of $H$) from the spectral parameter $\lambda$. 
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     author = {A. R. Its and V. B. Matveev},
     title = {Schr\"odinger operators with finite-gap spectrum and $N$-soliton solutions of the {Korteweg--de~Vries} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {51--68},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1975_23_1_a5/}
}
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A. R. Its; V. B. Matveev. Schr\"odinger operators with finite-gap spectrum and $N$-soliton solutions of the Korteweg--de~Vries equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 23 (1975) no. 1, pp. 51-68. http://geodesic.mathdoc.fr/item/TMF_1975_23_1_a5/