We consider wave propagation across an infinite waveguide of an arbitrary bounded cross-section, whose interior is blocked by two identical thick barriers with holes. When the holes are small, the waves over a broad range of frequencies are almost fully reflected. However, we show the existence of a resonance frequency at which the wave is almost fully transmitted, even for very small holes. This resonance scattering, which is known as tunneling effect in quantum mechanics, is demonstrated in a constructive way by rather elementary tools, in contrast to commonly used abstract methods such as searching for complex-valued poles of the scattering matrix or non-stationary scattering theory. In particular, we derived an explicit equation that determines the resonance frequency. The employed elementary tools make the paper accessible to non-experts and educationally appealing.
We consider a quantum theory of elastic light scattering from a macroscopic atomic sample existing in the Bose-Einstein condensate (BEC) phase. Following the second quantized formalism we introduce a set of coupled and closed diagram equations for the polariton propagator contributing to the T -matrix and scattering amplitude. Our approach allows us to follow important density corrections to the quasi-energy structure caused by static interaction and radiation losses associated with incoherent scattering in the case of near resonance excitation.
Abstract We study the resonant scattering for discrete time quantum walks on graphs with some tails. In our arguments, we reduce the study of resonances to the perturbation of eigenvalues of a finite rank matrix associated with the internal graph. Then we can apply Kato’s perturbation theory of matrices, and the reduction process of generalized eigenspaces allows us to derive an explicit asymptotic expansion of the scattering matrix. As a consequence, we obtain the resonant scattering at resonant energies.
In the present paper we shall be concerned with the stationary theory of scattering associated with the Schrodinger operator with a long-range potential. In quantum theory of scattering, many authors have investigated the existence and completeness of wave operators W = s-lim ee~y where J-»±oo HI and H2 are self-adjoint operators acting on a Hilbert space M. Among them, Kato and Kuroda gave an abstract time-independent approach to scattering theory. They derived a stationary form of the wave operator in the following way:
Most of the current theories on the
p-wave superfluid in cold atomic gases are based on the effective-range theory for the two-body scattering, where the low-energy
p-wave scattering amplitude
f1(k) is given by ${f}_{1}(k)=\ensuremath{-}1/[\mathit{ik}+1/(\mathcal{V}{k}^{2})+1/\mathcal{R}]$. Here
k is the incident momentum,
V and
R are the
k-independent scattering volume and effective range, respectively. However, due to the long-range nature of the van der Waals interaction between two colliding ultracold atoms, the
p-wave scattering amplitude of the two atoms is not described by the effective-range theory [J. Math. Phys. 4, 54 (1963); Phys. Rev. A 58, 4222 (1998)]. In this paper we provide an explicit calculation for the
p-wave scattering of two ultracold atoms near the
p-wave magnetic Feshbach resonance. We show that in this case the low-energy
p-wave scattering amplitude ${f}_{1}(k)=\ensuremath{-}1/[\mathit{ik}+1/({\mathcal{V}}^{\mathrm{eff}}{k}^{2})+1/({\mathcal{S}}^{\mathrm{eff}}k)+1/{\mathcal{R}}^{\mathrm{eff}}]$ where
Veff, Seff, and
Reff are
k-dependent parameters. Based on this result, we identify sufficient conditions for the effective-range theory to be a good approximation of the exact scattering amplitude. Using these conditions we show that the effective-range theory is a good approximation for the
p-wave scattering in the ultracold gases of
6Li and
40K when the scattering volume is enhanced by the resonance.