The Weyl algebra, bispectral operators and dynamics of poles in integrable systems
The author gives a survey on some recent results in the study of the bispectral problem. It is said that an ordinary differential operator \(L = L(x,\partial_x)\) is bispectral if it has an eigenfunction \(\psi(x,z)\) depending on the spectral parameter \(z\) which is also an eigenfunction of another differential operator \(A=A(z,\partial_z)\): \(L \psi = f(z) \psi\), \(A \psi = \theta(x) \psi\), where \(f\) and \(\theta\) are nonconstant functions. The author considers operators in the Weyl algebra generated by \(x\) and \(\partial_x\) with the standard commutation relation and, in particular, proves that an operator of nonzero order from this algebra is bispectral if and only if it belongs to the orbit of an operator with constant coefficients under the action of the group of automorphisms of the Weyl algebra. The author studies in detail an application of the Darboux transformation for constructing new bispectral operators from given bispectral operators. The text can serve as an introduction to the bispectral problem.
scientific article; zbMATH DE number 1979643