Transfer matrix for long-range potentials

Abstract

We extend the notion of the transfer matrix of potential scattering to a large class of long-range potentials $v(x)$ and derive its basic properties. We outline a dynamical formulation of the time-independent scattering theory for this class of potentials where we identify their transfer matrix with the $S$-matrix of a certain effective non-unitary two-level quantum system. For sufficiently large values of $|x|$, we express $v(x)$ as the sum of a short-range potential and an exactly solvable long-range potential. Using this result and the composition property of the transfer matrix, we outline an approximation scheme for solving the scattering problem for $v(x)$. To demonstrate the effectiveness of this scheme, we construct an exactly solvable long-range potential and compare the exact values of its reflection and transmission coefficients with those we obtain using our approximation scheme.

24 pages, accepted for publication in J. Phys. A

References in corpus (14)

• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• The transfer matrix: a geometrical perspective
• Wave reflection in dielectric media obeying spatial Kramers-Kronig relations
• Transfer Matrices as Non-Unitary S-Matrices, Multimode Unidirectional Invisibility, and Perturbative Inverse Scattering
• A Dynamical Formulation of One-Dimensional Scattering Theory and Its Applications in Optics
• Transfer Matrix Formulation of Scattering Theory in Two and Three Dimensions
• Bidirectional invisibility in Kramers-Kronig optical media
• Nonlinear Scattering and Its Transfer Matrix Formulation in One Dimension
• Transfer-matrix formulation of the scattering of electromagnetic waves and broadband invisibility in three dimensions
• Multi-Channel Transport in Disordered Medium under Generic Scattering Conditions
• Solving scattering problems in the half-line using methods developed for scattering in the full line
• Scattering Theory and $\mathcal{P}\mathcal{T}$-Symmetry

Cited by in corpus (7)

• Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions
• Transmission of low-energy scalar waves through a traversable wormhole
• Dynamical formulation of low-energy scattering in one dimension
• Scattering of TE and TM waves and quantum dynamics generated by non-Hermitian Hamiltonians
• A unified theory of tunneling times promoted by Ramsey clocks
• Tunneling Gravimetry
• Propagating-wave approximation in two-dimensional potential scattering