AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.
AbstractWe develop an accurate approximation of the normalized hyperbolic
operator sine family generated by a strongly positive operator
A in a Banach space X which represents the solution operator for the elliptic boundary
value problem. The solution of the corresponding inhomogeneous boundary value problem is found
through the solution operator and the Green function. Starting with the Dunford —
Cauchy representation for the normalized hyperbolic operator sine family and for the
Green function, we then discretize the integrals involved by the exponentially convergent
Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits
a two-level parallelism with respect to both the computation of resolvents and the
treatment of different values of the spatial variable x ∈ [0, 1].
We are concerned with the following nonlinear three-point fractional boundary value problem:D0+αut+λatft,ut=0,0<t<1,u0=0, andu1=βuη, where1<α≤2,0<β<1,0<η<1,D0+αis the standard Riemann-Liouville fractional derivative,at>0is continuous for0≤t≤1, andf≥0is continuous on0,1×0,∞. By using Krasnoesel'skii's fixed-point theorem and the corresponding Green function, we obtain some results for the existence of positive solutions. At the end of this paper, we give an example to illustrate our main results.
Inhomogeneous spectral moment sum rules for the retarded Green function and self-energy of strongly correlated electrons or ultracold fermionic atoms in optical lattices
