This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally...This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally, a short proof of a certain summarion formula is given展开更多
We review our calculation method, Gaussian expansion method （GEM）, to solve accurately the Schrodinger equations for bound, resonant and scattering states of few-body systems. Use is made of the Rayleigh-Ritz variat...We review our calculation method, Gaussian expansion method （GEM）, to solve accurately the Schrodinger equations for bound, resonant and scattering states of few-body systems. Use is made of the Rayleigh-Ritz variational method for bound states, the complex-scaling method for resonant states and the Kohn-type variational principle to S-matrix for scattering states. GEM was proposed 30 years ago and has been applied to a variety of subjects in few-body （3- to 5-body） systems, such as 1） few-nucleon systems, 2） few-body structure of hypernuelei, 3） clustering structure of light nuclei and unstable nuclei, 4） exotic atoms/molecules, 5） cold atoms, 6） nuclear astrophysics and 7） structure of exotic hadrons. Showing examples in our published papers, we explain i） high accuracy of GEM calculations and its reason, ii） wide applicability of GEM to various few-body systems, iii） successful predictions by GEM calculations before measurements. The total bound-state wave function is expanded in terms of few-body Gaussian basis functions spanned over all the sets of rearrangement Jacobi coordinates. Gaussians with ranges in geometric progression work very well both for short- range and long-range behavior of the few-body wave functions. Use of Gaussians with complex ranges gives much more accurate solution than in the case of real-range Gaussians, especially, when the wave function has many nodes （oscillations）. These basis functions can well be applied to calculations using the complex-scaling method for resonances. For the few-body scattering states, the amplitude of the interaction region is expanded in terms of those few-body Gaussian basis functions.展开更多
文摘This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally, a short proof of a certain summarion formula is given
文摘We review our calculation method, Gaussian expansion method （GEM）, to solve accurately the Schrodinger equations for bound, resonant and scattering states of few-body systems. Use is made of the Rayleigh-Ritz variational method for bound states, the complex-scaling method for resonant states and the Kohn-type variational principle to S-matrix for scattering states. GEM was proposed 30 years ago and has been applied to a variety of subjects in few-body （3- to 5-body） systems, such as 1） few-nucleon systems, 2） few-body structure of hypernuelei, 3） clustering structure of light nuclei and unstable nuclei, 4） exotic atoms/molecules, 5） cold atoms, 6） nuclear astrophysics and 7） structure of exotic hadrons. Showing examples in our published papers, we explain i） high accuracy of GEM calculations and its reason, ii） wide applicability of GEM to various few-body systems, iii） successful predictions by GEM calculations before measurements. The total bound-state wave function is expanded in terms of few-body Gaussian basis functions spanned over all the sets of rearrangement Jacobi coordinates. Gaussians with ranges in geometric progression work very well both for short- range and long-range behavior of the few-body wave functions. Use of Gaussians with complex ranges gives much more accurate solution than in the case of real-range Gaussians, especially, when the wave function has many nodes （oscillations）. These basis functions can well be applied to calculations using the complex-scaling method for resonances. For the few-body scattering states, the amplitude of the interaction region is expanded in terms of those few-body Gaussian basis functions.
