On the Toda Lattice. II: Inverse-Scattering Solution

The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations

“…We follow [13] (see also [35]), where the theory summarized below can be found. One considers in the Hilbert space l 2 of square-summable, doubly infinite complex sequences {f (n); −∞ < n < ∞}, the following eigenvalue problem…”

“…Here a n > 0 for all n ∈ Z; it is assumed that a n → 1/2 and b n → 0 rapidly (exponentially, for example) as |n| → ∞, see [13] for the precise assumptions.…”

“…L − ) denote the strictly upper (resp. lower) part of L. When flowing, the operator L(s 1 ) stays reflectionless, with the same eigenvalues λ j = (z j + z −1 j )/2 and norming constants c j (s 1 ) = c j exp s 1 (z j − z −1 j )/2 , see [13]. When the c j 's are allowed to flow according to (2.17), one gets a solution of the full Toda lattice hierarchy.…”

“…In this paper, I propose to further explore the "finite number of terms phenomenon" found in [16,19] by studying the case when L in (1.2) is a doubly infinite Jacobi matrix, leading (when taken as initial condition) to a soliton solution of the Toda lattice equations [13,35]. For sake of brevity, I shall call these Jacobi matrices "Toda solitons".…”

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“…admits infinite conserved quantities, can be solved for rapidly decreasing boundary conditions through the method of inverse scattering [3] and admits explicit quasi-periodic solutions by algebro-geometric methods [4]- [6]. It has found important applications in many different fields, in particular in the theory of Gromov-Witten invariants of CP 1 , where the present extension plays a particular role [7].…”