Complex path integral representation for semiclassical linear functionals
Let \(L\) be a complex valued linear functional defined on all polynomials. \(L\) is determined by its moments \[ \langle L,z^j\rangle:= L(z^j):= \mu_j,\quad j\geq 0. \] In this interesting paper, the authors study such linear functionals that are determined by a distributional equation of the form \[ D(\phi L)+ \psi L= 0,\tag{1} \] where \(\phi\) and \(\psi\) are polynomials with \(\text{deg}(\psi)\geq 1\). This equation may be interpreted in the usual distributional way: \(D\) denotes the distributional derivative, defined by \(\langle DL,p\rangle=- \langle L,p'\rangle=- L(p')\) for all polynomials \(p\), and \(\langle\psi L,p\rangle=\langle L,\psi p\rangle= L(\psi p)\). Thus (1) becomes \[ -\phi L(p')+ L(\psi p)= 0, \] for all polynomials \(p\). A linear functional satisfying (1) is called semiclassical. (The name arises from the connection with semiclassical orthogonal polynomials, which provide a strong motivation for this study.) In an earlier paper, the authors considered the case \(\text{deg}(\phi)> \text{deg}(\psi)\). Here, they consider the case \(\text{deg}(\phi)\leq \text{deg}(\psi)\). They establish a representation \(\langle L,p\rangle= \int_\gamma (pw)(z)dz\), where \(\gamma\) is a suitable contour, and \(w\) is a solution of the differential equation \((\phi w)'+ \psi w=0\). The description of \(w\) and \(\gamma\) is given in terms of more detailed assumptions about \(\phi\) and \(\psi\).
scientific article; zbMATH DE number 1214034