Among all of the non-Hermitian large-tridiagonal-matrix quantum Hamiltonians we choose a subclass with the structure resembling the ``benchmark'' realistic Bose-Hubbard model. We demonstrate that this choice can be declared user-friendly in the sense that the underlying singular values can be specified via a ``Hermitized'' Schrödinger-like equation. In particular, the related ``Hermitized'' Green's functions is shown given the two alternative compact and numerically efficient matrix continued fraction forms.
A strong laser field interacting with an atom or a molecule can induce a coherent process which includes a large number of energy levels. If we can find a transformation which eliminates the explicit time dependence of the interaction Hamiltonian (for example, the rotating-wave approximation), we can reduce the dynamic problem to the consideration of eigenvalues and eigenvectors of the Hamiltonian. However, the complexity of an arbitrary multilevel Hamiltonian usually forces us to use numerical methods, and, therefore, not much general understanding about multilevel systems can be achieved. We consider the multilevel problems in a more general setting. Our approach is based on the observation that an arbitrary Hamiltonian matrix can be transformed into a tridiagonal form by unitary transformation.1 A tridiagonal form represents a ladder-type multilevel system. From all W-level systems the ladder configuration is the simplest in the sense that the number of level connections is the smallest possible. This means that the most effective analysis of a given multilevel system can be achieved using the transformation to ladder configuration. The spectral analysis of ladders is facilitated by the connections to orthogonal polynomials and continued fractions. The investigations about the spectral properties of ladders are reported.
Accurate extraction of linearized quantum circuit models from electromagnetic simulations is essential for the design of superconducting circuits. We present a quantization framework based on the driving-point admittance
Yin(s) seen by a Josephson junction embedded in an arbitrary passive linear environment. By taking the Schur complement of the nodal admittance matrix, we show that the linearized coupled system obeys an eigenvalue-dependent boundary condition,
sYin(s)+1/LJ=0, whose roots determine the dressed linear mode frequencies. This boundary condition admits an exact continued fraction representation: any positive-real admittance can be realized as a canonical Cauer ladder, yielding a tridiagonal (Jacobi) structure that enables certified convergence bounds via interlacing theorems.For the full nonlinear Hamiltonian, we treat Josephson junctions in the charge basis, where each cosine potential is exactly tridiagonal, and couple them to cavity modes in the Fock basis; in the general multi-junction case this yields a block-tridiagonal structure solvable by matrix continued fractions, enabling systematic diagonalization across all coupling regimes from dispersive through ultrastrong and deep strong coupling. The resulting quantization procedure is: (i)~compute or measure
Yin(s), (ii)~solve the boundary condition to obtain dressed eigenfrequencies, (iii)~synthesize an equivalent passive network, and (iv)~quantize while retaining the full cosine nonlinearity of the Josephson junction. We prove that junction participation decays as
O(ωn−1) at high frequencies for any circuit with finite shunt capacitance, ensuring ultraviolet convergence of perturbative corrections without imposed cutoffs.