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# The integration of double-infinite Toda lattice by means of inverse spectral problem and related questions

### Abstract

The solution of the Cauchy problem for differential-difference double-infinite Toda lattice by means of inverse spectral problem for semi-infinite block Jacobi matrix is given. Namely, we construct a simple linear system of three differential equations of first order whose solution gives the spectral matrix measure of the aforementioned Jacobi matrix. The solution of the Cauchy problem for the Toda lattice is given by the procedure of orthogonalization w.r.t. this spectral measure, i.e. by the solution of the inverse spectral problem for this Jacobi matrix.

Key words: Toda lattice, Cauchy problem, Jacobi matrix, direct and inverse spectral problems, generalized eigenvectors expansion.

### Article Information

 Title The integration of double-infinite Toda lattice by means of inverse spectral problem and related questions Source Methods Funct. Anal. Topology, Vol. 15 (2009), no. 2, 101-136 MathSciNet MR2553529 Copyright The Author(s) 2009 (CC BY-SA)

### Authors Information

Yurij Berezansky
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

### Citation Example

Yurij Berezansky, The integration of double-infinite Toda lattice by means of inverse spectral problem and related questions, Methods Funct. Anal. Topology 15 (2009), no. 2, 101-136.

### BibTex

@article {MFAT512,
AUTHOR = {Berezansky, Yurij},
TITLE = {The integration of double-infinite Toda lattice by means of inverse spectral problem and related questions},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {15},
YEAR = {2009},
NUMBER = {2},
PAGES = {101-136},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=512},
}