Linearization of double-infinite Toda lattice by means of inverse spectral problem (Q2896660)

The author gives a detailed rigorous exposition of his spectral approach to the integration of the Cauchy problem for the double-infinite Toda lattice; for a shorter description of this method see \textit{Y. Berezansky} [Methods Funct. Anal. Topol. 15, No. 2, 101--136 (2009; Zbl 1199.34026)]. This includes a completely rigorous exposition of some results regarding the semi-infinite Toda lattice given earlier on a formal level (see, in particular, \textit{Y. M. Berezanski} [Rep. Math. Phys. 24, No. 1, 21--47 (1986; Zbl 0652.35098)]). Some applications deal with the Cauchy problem and the shock problem for the equation NEWLINE\[NEWLINE \ddot{x}_n(t)=e^{x_{n-1}(t)-x_n(t)}-e^{x_n(t)-x_{n+1}(t)}, NEWLINE\]NEWLINE where \( n\in\mathbb Z,t\in [0,T)\).