Unitary transform diagonalizing the Confluent Hypergeometric kernel

TL;DR

This work diagonalizes the determinantal point process induced by the confluent hypergeometric kernel $K^s$ via the explicit unitary transform $\mathcal{T}_s$. It develops a Paley-Wiener framework for $\mathcal{T}_s$, proves unitarity, and transfers Wiener–Hopf factorization to the $G_f$-picture through a unitary equivalence with the classical operator $W_f$. A hierarchical decomposition of the Paley-Wiener space $\mathcal{PW}_s$ into one-dimensional components $L^{(s,n)}$ is given, along with explicit bases $\mathcal{L}_{(s,n)}$ and the relation $\mathcal{T}_s^*\mathbb{I}_{[0,1]}\mathcal{T}_s = \psi K^s \psi^*$. These results yield concrete factorization properties and trace formulas, connecting to the Palm hierarchy and sine-process limits in determinantal point processes.

Abstract

We consider the image of the operator, inducing the determinantal point process with the confluent hypergeometric kernel. The space is described as the image of $L_2[0, 1]$ under a unitary transform, which generalizes the Fourier transform. For the derived transform we prove a counterpart of the Paley-Wiener theorem. We use the theorem to prove that the corresponding analogue of the Wiener-Hopf operator is a unitary equivalent of the usual Wiener-Hopf operator, which implies that it shares the same factorization properties and Widom's trace formula. Finally, using the introduced transform we give explicit formulae for the hierarchical decomposition of the image of the operator, induced by the confluent hypergeometric kernel.

Theorems & Definitions (30)

• Theorem 1.1
• Remark
• Remark
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 2.1: Bourgade, Nikeghbali, Rouault BNR_06
• Remark
• Lemma 2.2
• proof : Proof of the identity \ref{['2_eq:CD_cont']}