On the exact WKB analysis of higher order simple-pole type operators
The authors recently introduced a class of higher-order simple-pole type linear differential operators with a large parameter \(\eta\) \[ P(x,d_x,\eta)=d_x^m+(\eta A_{1,0}(x)+x^{-1}A_{1}(x,\eta))\,d_x^{m-1}+x^{-1}\sum_{j=0}^{m-2}A_{j}(x,\eta)\,d_x^j, \] where \(A_{j}(x,\eta)\) are formal series in \(\eta^{-1}\) with coefficients holomorphic in \(x\). In the present paper, they prove a decomposition theorem for this operator \(P\), namely, \(xP=Q^{(1)}\cdots Q^{(m-2)}xR\), where \(Q^{(l)}=d_x-\eta q^{(l)}(x,\eta)\) are first order operators and \(R=R(x,d_x,\eta)\) is a second order simple-pole type differential operator which is uniquely determined by \(P(x,d_x,\eta)\) regardless the choice of \(Q^{(l)}\). Thus, from the point of view of exact WKB analysis, the Stokes geometry of any higher-order simple-pole type operator reduces in essence to the study of a Schrödinger operator.
scientific article; zbMATH DE number 5941206
scientific article