One-point commuting difference operators of rank one

摘要

We consider one-point commuting difference operators of rank one. Thecoefficients of these operators depend on a functional parameter, shiftoperators being included only with positive degrees. We study these operatorsin the case of hyperelliptic spectral curve when the marked point coincideswith the branch point. We construct examples of operators with polynomial andtrigonometric coefficients. Moreover, difference operators with polynomialcoefficients can be embedded in the differential ones with polynomialcoefficients. This construction provides a new way of constructing commutativesubalgebras in the first Weyl algebra.