Spectral theory of the Nazarov-Sklyanin Lax operator
Let \(F=\mathbb{C}[V_1, V_2,\ldots]\) be a graded polynomial ring, such that \(\deg V_k=k\). For any \(\hbar\in \mathbb{C}\), \(V_{-k}\) denotes the differential operator in \(F\) defined by \(V_{-k}=\hbar k\frac{\partial}{\partial V_k}\). Taking into account that \([V_{-k}, V_k]=\hbar k\), then \(F=\mathbb{C}[V_1, V_2,\ldots]\) is the Fock space representation of the Heisenberg algebra at level \(\hbar\), that is identified with the ring of symmetric functions. A new variable \(w\) with \(\deg w=1\) is introduced and the graded polynomial and Laurent polynomial rings \(F[w]=\mathbb{C}[w,V_1, V_2,\ldots]\), \(F[w,w^{-1}]=\mathbb{C}[w, w^{-1},V_1, V_2,\ldots]\) are considered. Let the projection \(\pi: F[w,w^{-1}]\rightarrow F[w]\) be the linear extension of \(\pi w^l=\begin{cases} w^l & \mbox{ if } l\geq 0;\\ 0 & \mbox{ if } l< 0 \end{cases}\). For any \(\bar{\epsilon}, \hbar\in\mathbb{C}\) and \(\partial_w=\frac{\partial}{\partial w}\), the Nazarov-Sklyanin Lax operator \(\mathcal{L}\) is the graded linear operator \(\mathcal{L}: F[w]\rightarrow F[w]\) in the graded polynomial ring \(F[w]=\mathbb{C}[w,V_1, V_2,\ldots]\) defined by \[\mathcal{L}=\bar{\epsilon}w\partial_w+\sum_{k=1}^\infty V_{-k}w^k+\sum_{k=1}^\infty V_{k}\pi w^{-k}\] for \(V_{-k}\) and \(\pi\) defined above. The main results are in Theorem 1.4, in which the spectrum of \(\mathcal{L}\) in \(F[w]\) is determined and a distinguished polynomial basis \(\psi^s_\lambda\) of eigenfunctions of \(\mathcal{L}\) satisfying \(\pi_0 \psi^s_\lambda=j_\lambda\) is identified. Here \(\pi_0: F[w]\rightarrow F\) is the projection defined by the setting \(w=0\) and \(j_\lambda\) are Jack polynomials.
scientific article; zbMATH DE number 7757105
scientific article