This paper investigates the spectrum of the \(\bar{\partial}\)-Laplace operator on zero-forms for the quantum quadric \(\mathcal O_q(\mathcal Q_N)\), based on the existence of the Heckenberger-Kolb differential calculus of irreducible quantum flag manifolds of types \(B_n\) and \(D_n\) in the list of quantum homogeneous spaces (Table 1,§2.3, page 5; Table 2, Appendix A, page 18).
The main result is Proposition 4.3, stating that for a real quantum parameter \(q\in (0,+\infty)\setminus \{1\}\) sufficiently close to \(1\) and the Hilbert space completion \(L^2(\Omega^0)\) of the *-algebra (Definition 2.1) of \[\Omega^0=B, \Omega^k = \bigoplus_{a+b=k} \Omega^{(a,b)}, {\Omega^{(a,b)}}^*= \Omega^{(b,a)} \] with respect to the inner product (Definition 2.2) \[ \langle .,.\rangle : \Omega^{(\bullet,\bullet)} \times \Omega^{(\bullet,\bullet)} \to \mathbb C; (\omega, \nu) = \mathbf h\circ g_\sigma(\omega, \nu), \] the eigenvalue of the Dolbeault Laplace operator \(D : L^2({\Omega^0}) \to L^2({\Omega^0})\) with \(D(\Delta_{\bar{\partial}}) = L^2(\Omega^0)\) goes to infinity with finite multiplicity and it therefore has compact resolvent.