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**Singular solitons and indefinite metrics.**
*(English.
Russian original)*
Zbl 1248.34132

Dokl. Math. 83, No. 1, 56-58 (2011); translation from Dokl. Akad. Nauk 436, No. 3, 302-305 (2011).

It is well known that the differential equation \(L \Psi := - \Psi'' + u(x) \Psi = \lambda \Psi\) with \(\lambda \in {\mathbb C}\) leads to eigenfunction expansion theorems if \(u\) is an \(L^1\)-function. The situation is more involved if the real potential \(u\) has singularities. The present paper presents a function theoretic approach to potentials \(u\) which are \(C^\infty\)-smooth outside isolated singularities \(x_j\) and meromorphic in a neighbourhood of \(x_j\).

Two different types of potentials are considered: a) singular finite-gap potentials when \(u\) is periodic (i.e. \(u(x + T) = u(x)\)) and b) singular solitons when \(u(x) \rightarrow 0\) as \(|x| \rightarrow \infty\). The inner product \(\langle \Psi_1, \Psi_2 \rangle = \int_0^T \Psi_1(x) \overline{\Psi_2}(\overline{x}) \, dx\) in case a) and \(\langle \Psi_1, \Psi_2 \rangle = \int_{-\infty}^{\infty} \Psi_1(x) \overline{\Psi_2}(\overline{x}) \, dx\) in case b) is considered on the space \({\mathcal F}_{(X)}\) of \(C^\infty\)-smooth functions outside singularities in \(x_j\). It is observed that it has finitely many negative squares and its number is calculated. However, the associated Pontrjagin space topology seems to be not used in the following.

Two expansion theorems for \({\mathcal F}_{(X)}\) are obtained with a function representation in terms of a series in case a) and in terms of an integral plus finitely many terms in case b). It is mentioned that a certain representation of a rational solution of the Korteweg-de Vries equation \(u_t = 6u u_x - u_{xxx}\) plays an important role in the proofs of the theorems which are not presented in the paper.

Two different types of potentials are considered: a) singular finite-gap potentials when \(u\) is periodic (i.e. \(u(x + T) = u(x)\)) and b) singular solitons when \(u(x) \rightarrow 0\) as \(|x| \rightarrow \infty\). The inner product \(\langle \Psi_1, \Psi_2 \rangle = \int_0^T \Psi_1(x) \overline{\Psi_2}(\overline{x}) \, dx\) in case a) and \(\langle \Psi_1, \Psi_2 \rangle = \int_{-\infty}^{\infty} \Psi_1(x) \overline{\Psi_2}(\overline{x}) \, dx\) in case b) is considered on the space \({\mathcal F}_{(X)}\) of \(C^\infty\)-smooth functions outside singularities in \(x_j\). It is observed that it has finitely many negative squares and its number is calculated. However, the associated Pontrjagin space topology seems to be not used in the following.

Two expansion theorems for \({\mathcal F}_{(X)}\) are obtained with a function representation in terms of a series in case a) and in terms of an integral plus finitely many terms in case b). It is mentioned that a certain representation of a rational solution of the Korteweg-de Vries equation \(u_t = 6u u_x - u_{xxx}\) plays an important role in the proofs of the theorems which are not presented in the paper.

Reviewer: Andreas Fleige (Dortmund)

### MSC:

34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |

35Q51 | Soliton equations |

46C20 | Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) |

35Q53 | KdV equations (Korteweg-de Vries equations) |

### Keywords:

singular solitons; singular finite-gap potentials; indefinite metric; eigenfunction expansion; Korteweg-de Vries equation
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\textit{P. G. Grinevich} and \textit{S. P. Novikov}, Dokl. Math. 83, No. 1, 56--58 (2011; Zbl 1248.34132); translation from Dokl. Akad. Nauk 436, No. 3, 302--305 (2011)

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### References:

[1] | P. G. Grinevich and S. P. Novikov, Usp. Mat. Nauk 64(4), 45–72 (2009). |

[2] | V. A. Arkad’ev, A. G. Pogrebkov, and M. K. Polivanov, Zap. Nauchn. Sem. LOMI 133, 17–37 (1984). |

[3] | I. M. Krichever, Usp. Mat. Nauk 44(2), 121–184 (1989). |

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