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Article

An hbar-expansion of the Toda hierarchy

Analysis and Mathematical Physics. 2012. No. 2. P. 171-214.
Takasaki K., Takebe T.

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 (L_, M_) of Lax and Orlov-Schulman operators. This Riemann–Hilbert problem is translated to the language of the dressing operators W and W_. The dressing operators are set in an exponential form as W=exp X/hbar and W_=exp phi/hbar exp X_/hbar, and the auxiliary operators X, X_ and the function phi are assumed to have hbar-expansions X=X0+ hbar X1+…, X=X0_+hbar X1_+… and phi=phi0+hbar phi1 +…. The coefficients of these expansions turn out to satisfy a set of recursion relations. X, X_ and phi are recursively determined by these relations. Moreover, the associated wave functions are shown to have the WKB form Psi = exp(S/hbar) and Psi_=exp(S_/hbar), which leads to an hbar-expansion of the logarithm of the tau function.