45 research outputs found
Characterizing Multiple Solutions to the Time - Energy Canonical Commutation Relation via Internal Symmetries
Internal symmetries can be used to classify multiple solutions to the time
energy canonical commutation relation (TE-CCR). The dynamical behavior of
solutions to the TE-CCR posessing particular internal symmetries involving time
reversal differ significantly from solutions to the TE-CCR without those
particular symmetries, implying a connection between the internal symmetries of
a quantum system, its internal unitary dynamics, and the TE-CCR.Comment: Accepted for publication in Physical Review A, 10 page
Regularized Limit, analytic continuation and finite-part integration
Finite-part integration is a recent method of evaluating a convergent
integral in terms of the finite-parts of divergent integrals deliberately
induced from the convergent integral itself [E. A. Galapon, Proc. R. Soc., A
473, 20160567 (2017)]. Within the context of finite-part integration of the
Stieltjes transform of functions with logarithmic growths at the origin, the
relationship is established between the analytic continuation of the Mellin
transform and the finite-part of the resulting divergent integral when the
Mellin integral is extended beyond its strip of analyticity. It is settled that
the analytic continuation and the finite-part integral coincide at the regular
points of the analytic continuation. To establish the connection between the
two at the isolated singularities of the analytic continuation, the concept of
regularized limit is introduced to replace the usual concept of limit due to
Cauchy when the later leads to a division by zero. It is then shown that the
regularized limit of the analytic continuation at its isolated singularities
equals the finite-part integrals at the singularities themselves. The treatment
gives the exact evaluation of the Stieltjes transform in terms of finite-part
integrals and yields the dominant asymptotic behavior of the transform for
arbitrarily small values of the parameter in the presence of arbitrary
logarithmic singularities at the origin