Spectrum of the d-bar Neumann Laplacian on the Fock space

TL;DR

This paper explicitly computes the spectrum of the d-bar-Neumann Laplacian on the Fock space, revealing positive integer eigenvalues with infinite multiplicity, and links it to magnetic Schrödinger operators and the Witten-Laplacian.

Contribution

It provides an explicit spectral analysis of the d-bar-Neumann Laplacian on the Fock space, connecting it to well-studied operators in mathematical physics.

Findings

Spectrum consists of positive integers with infinite multiplicity.

Spectral analysis relates to Schrödinger operators with magnetic fields.

Connections to the complex Witten-Laplacian are established.

Abstract

The spectrum of the d-bar-Neumann Laplacian on the Fock space $L^2(\mathbb{C}^n, e^{-|z|^2})$ is explicitly computed. It turns out that it consists of positive integer eigenvalues each of which is of infinite multiplicity. Spectral analysis of the d-bar-Neumann Laplacian on the Fock space is closely related to Schr\"odinger operators with magnetic field and to the complex Witten-Laplacian.