Inflationary power spectrum from the Lanczos algorithm

The generalized Lanczos algorithm can provide a universal method for constructing a wave function under the the Hamiltonian group structure. Based on this fact, we obtain an open two-mode squeezed state as the quantum origin for the curvature perturbation. In light of this wave function in the open system, we successfully develop a new method to calculate its corresponding power spectrum by using the Bogoliubov transformation. Unlike traditional approaches, we explicitly retain the Bogoliubov coefficients in terms of the squeezing amplitude \( r_k \) and the squeezing rotation angle \( \phi _k \). As a result, the power spectrum of the open two-mode squeezed state will match that of the Bunch–Davies vacuum numerically. Furthermore, the derivation of the open two-mode squeezed state relies on a Meixner polynomial of the second kind (equivalent to the generalized Lanczos algorithm) and the symmetry of the Hamiltonian. Therefore, our research may offer a new insight into the calculation of correlation functions through a group-theoretic perspective.