Commuting elements in the first Weyl algebra over \(\mathbb{Q} \)
The aim of this paper is to study the commuting differential operators whose coefficients are polynomials over \(\mathbb{Q}\) and \(\mathbb{Z}\), i.e., the commuting elements of the first Weyl algebra over \(\mathbb{Q}\) and \(\mathbb{Z}\), as well as the actions of the groups Aut(\(A_1(\mathbb{Q})\)) and Aut(\(A_1(\mathbb{Z})\)) on the solution sets to some equations, where \(A_1(K)=K[x][\partial_x]\) is the first Weyl algebra over \(K\), with \(K=\mathbb{C}\), \(\mathbb{Q}\), and \(\mathbb{Z}\). The author considers the commuting self adjoint differential operators of rank \(2\) and orders \(4\) and \(6\) with the following elliptic spectral curve \[ w^2=z^3+c_2z^2+c_1z+c_0, \] and the Krichever-Novikov formally self adjoint operators \[ L_4==(\partial^2_x+V(x))^2+W(x), \] \[ L_6=(\partial^2_x+V(x))^3+\frac{1}{2}(c_2+3W(x))(\partial^2_x+V(x))+\frac{3}{2}W_x(x)\partial_x+\frac{5}{4}W_{xx}(x), \] where \(V(x)\) and \(W(x)\) are rational functions. The automorphism group of the first Weyl algebra over \(\mathbb{Q}\) acts on the commuting differential operators with polynomial coefficients over \(\mathbb{Q}\). The author shows that the orbit set is infinite for a fixed elliptic spectral curve over \(\mathbb{Q}\) with at least one rational point. The main result obtained in this paper can be formulated as follows: suppose that the above elliptic spectral curve with rational coefficients \(c_i\in\mathbb{Q}\) contains at least one point \((z_0,w_0)\) with rational coordinates, then there is a countable family \(\{(L_{4,\alpha},L_{6,\alpha}\in\mathbb{Q}[x][\partial_x]\}\) of commuting differential operators parametrized by rationals \(\alpha\in\mathbb{Q}\) with this spectral curve such that the pairs of operators \((L_{4,\alpha},L_{6,\alpha})\) and \((L_{4,\alpha'},L_{6,\alpha'})\) for distinct \(\alpha,\alpha'\in\mathbb{Q}\) cannot be taken into each other by the elements of Aut(\(A_1(\mathbb{Q})\)). It is also deduced that the orbit set of the action of Aut(\(A_1(\mathbb{Q})\)) on the solution set to the above elliptic spectral curve is infinite. This paper is organized as follows: Section 1 deals with the case that \(W(x)\) is a polynomial of degree \(1\). The author shows that each rational point on the elliptic spectral curve corresponds to infinitely many orbits. The main result follow from the arguments of this Section. Section 2 deals with the case that \(W(x)\) is a polynomial of degree \(2\). It shown that each rational but not branch point on the spectral curve corresponds to its own orbit. Section 3 is devoted to Aut(\(A_1(\mathbb{Z})\)). The author shows that for the operators under consideration the number of orbits is finite.
polynomial coefficients over \(\mathbb{Q} \)
scientific article; zbMATH DE number 7596649
scientific article