non-Hermitian Eigenvalue Problem Research Articles

Gamow vectors as solutions of a non-hermitian eigenvalue problem

Gamow vectors have been extensively used for describing resonance phenomena, but they do not belong to the usual Hilbert space. They can be obtained for a single channel potential by solving the Schrödinger equation with out-going boundary conditions. A new method for calculating Gamow vectors through the solution of a matrix eigenvalue problem is presented. The matrix to be diagonalized is real and non-symmetric, allowing the treatment of bound states and resonances on the same footing. The advantage of the method is to be equally applicable to the coupled channel problem when the standard procedure of solving the set of radial differential equations have the problem of denning the regular solution at the origin. Furthermore, the solution of the eigenvalue problem gives directly the Gamow momenta and vectors, thus avoiding the usual search in the complex momentum plane which can be rather cumbersome. Applications to single as well as coupled channel cases, both for model potentials and realistic nucleon-nucleon interaction, show great accuracy of the method.

Floquet-Liouville supermatrix approach: Time development of density-matrix operator and multiphoton resonance fluorescence spectra in intense laser fields.

A Floquet-Liouville supermatrix (FLSM) approach is presented for nonperturbative treatment of the time development of the density-matrix operator of atoms and molecules exposed to intense polychromatic fields. By extending the many-mode Floquet theory (MMFT) recently developed, the time-dependent Liouville equation for the density matrix of quantum systems undergoing relaxations (due to radiative decays and collisional dampings, etc.) can be transformed into an equivalent time-independent non-Hermitian FLSM eigenvalue problem. This yields a numerically stable and computationally efficient approach for the unified treatment of nonresonant and resonant, one- and multiple-photon, steady-state and transient phenomena in nonlinear optical processes, much beyond the conventional rotating-wave-approximation (RWA) method. Connections of the FLSM approach to the MMFT in the limit of zero relaxations are also made, providing the understanding of the physical significance of FLSM supereigenvalues and eigenvectors. In addition to the exact FLSM formalism, we have also presented higher-order perturbative results, based on the extension of the generalized Van Vleck (GVV) nearly degenerate perturbation theory, appropriate for somewhat weaker fields and near-resonant processes, but beyond the RWA limit. The implementation of the GVV method in the time-independent Floquet-Liouvillian allows the reduction of the infinite-dimensional FLSM into a finite-dimensional GVV-Liouville matrix, from which essential analytical results are readily obtained. As an illustration of the usefulness of the new formalism, we extend both the FLSM and the GVV methods to a formal study of the multiphoton-induced resonance fluorescence spectra of two-level systems subject to purely radiation relaxations. Both the time-averaged power spectrum and the time-dependent physical spectrum are exploited in details, and novel new features in intense fields are pointed out.

Description of uniform many-particle systems in density operator function space

Assuming the ground state wavefunction, ψ 0, of a boson fluid is known, and writing the excited state wavefunctions in the form Fψ 0, a linear eigenvalue equation of the form HF = EF is obtained, where E 0 + E is the excited state energy, E 0 is the ground state energy, and H is a non-hermitian operator which depends in a simple way upon U ≡ ln ψ 0 2 instead of the potential energy function. An extremum principle is derived in terms of an auxiliary hermitian Hamiltonian operator, H′. The many-body boson plane-wave basis, ϱ n (k 1 … k n ) is used to express U in terms of its Fourier components (ordered conveniently in terms of the number of nonzero arguments), making it possible to calculate matrix elements of ovcirc| H and H′ in that basis. A perturbation theory similar to Brillouin-Wigner perturbation theory is developed for the non-hermitian eigenvalue problem. Nonorthogonal perturbation theory is developed for the correlated basis ϱ n ψ 0. The requirement that these two perturbation theories be consistent produces useful relationships between the components of U and the static structure functions of ψ 0. These relationships are shown to reduce to previous results in the extreme case of low density and weak interactions.

On the correct expansion of a Green function into a set of eigenfunctions connected with a non-Hermitian eigenvalue problem considered by Morse

It is shown that the analysis of Morse of a non-Hermitian eigenvalue problem, connected with a string with non-rigid supports, leads to an erroneous expansion of the Green function into a set of related eigenfunctions. The correct expansion is derived.

Linewidth of a detuned single mode laser near threshold

The linewidth of a detuned single mode laser is calculated. Using the Fokker-Planck equation method, the problem is reduced to a nonhermitian eigenvalue problem. Numerical results near threshold and an approximate linewidth formula are given.

Two-particle Green functions and nuclear structure

The two-particle Green function is studied in terms of the ladder approximation and applied to the two-particle shell model. The resulting non-Hermitian eigenvalue problem is solved in a simple schematic model, and conclusions are drawn regarding electromagnetic transition probabilities and the cross sections for two-particle transfer reactions.