Spectrum of the $\bar{\partial}$-Laplace operator on zero forms for the quantum quadric $\mathcal{O}_q(\textbf{Q}_N)$

TL;DR

This paper investigates the spectral properties of the $ar{ ext{d}}$-Laplacian on zero forms for quantum quadrics, revealing that eigenvalues grow unbounded with finite multiplicity, advancing understanding of quantum geometric operators.

Contribution

It provides a detailed spectral analysis of the $ar{ ext{d}}$-Laplacian on quantum quadrics, a novel study in the context of quantum flag manifolds of types B and D.

Findings

Eigenvalues tend to infinity

Eigenvalues have finite multiplicity

Spectral properties mirror classical Laplacian behavior

Abstract

We study the Laplacian operator $\Delta_{\bar{\partial}}$ associated to a K\"ahler structure $(\Omega^{(\bullet, \bullet)}, \kappa)$ for the Heckenberger--Kolb differential calculus of the quantum quadrics $\mathcal{O}_q(\textbf{Q}_N)$, which is to say, the irreducible quantum flag manifolds of types $B_n$ and $D_n$. We show that the eigenvalues of $\Delta_{\bar{\partial}}$ on zero forms tend to infinity and have finite multiplicity.