Second-Order Semiregular Non-Commutative Harmonic Oscillators: The Spectral Zeta Function

We study the properties of the spectral zeta function associated to a class of elliptic semiregular differential systems, including models of semiregular NCHOs, relevant to Quantum Optics. By “semiregular syste” we mean a pseudodifferential systems with a step \(-j\) in the homogeneity of the jth-term in the asymptotic expansion of the symbol. More precisely, we state a theorem about the continuation of the spectral zeta function of a semiregular elliptic differential system \(A^{\mathrm {w}}=\left (A^{\mathrm {w}}\right )^{*}>0\) as a function whose poles set has only one accumulation point that is the negative infinity.