An ${\hbar}$-expansion of the Toda hierarchy

Abstract

A construction of general solutions of the ${\hbar}$ -dependent Toda hierarchy is presented. The construction is based on a Riemann–Hilbert problem for the pairs (L, M) and ${(\bar {L},\bar {M})}$ of Lax and Orlov-Schulman operators. This Riemann–Hilbert problem is translated to the language of the dressing operators W and $\bar {W}$ . The dressing operators are set in an exponential form as ${W = e^{X/\hbar}}$ and ${\bar {W} = e^{\phi/\hbar}e^{\bar {X}/\hbar}}$ , and the auxiliary operators ${X, \bar {X}}$ and the function ${\phi}$ are assumed to have ${\hbar}$ -expansions ${X = X_0 + \hbar X_1 + . . . , \bar {X}= \bar {X}_0 + \hbar\bar {X}_1 + . . .}$ and ${\phi = \phi_0 + \hbar\phi_1 + . . .}$ . The coefficients of these expansions turn out to satisfy a set of recursion relations. ${X, \bar {X}}$ and ${\phi}$ are recursively determined by these relations. Moreover, the associated wave functions are shown to have the WKB form ${\Psi = e^{S/\hbar}}$ and ${\bar {\Psi}= e^{\bar {S}/\hbar}}$ , which leads to an ${\hbar}$ -expansion of the logarithm of the tau function.