Poles of the S-matrix for a complex potential

Colloquium: Physics of the Riemann hypothesis

“…Joffily introduces a mapping between these zeros of f + (k) onto the critical line and shows they coincide with the non-trivial zeros of the Riemann zeta function. He associates this artificial system with a vacuum and the zeros are interpreted as an infinity of virtual resonances, and thus reflect the chaotic nature of the vacuum (Joffily, 2003(Joffily, , 2004. This interpretation has also been extended using relativistic scattering (Joffily, 2007).…”

“…Joffily, motivated by Pavlov and Fadeev (Pavlov and Fadeev, 1975), examined the scattering states of a non-relativistic, spinless particle under the influence of a spherically symmetric, local and finite potential. He examined the Jost solutions of this scattering problem (Joffily, 2003), which differ from the physical solution of the Schrödinger equation in their asymptotics 8 (Alfaro and Regge, 1965;Newton, 1982). In standard non-relativistic scattering theory the S-matrix is given by…”

Complex-scaled spectrum completeness for pedestrians

“…This scheme of proofs was made possible by the consideration of a slightly modified CSM Hamiltonian, as shown by Eq.(1). Locations of zeroes of the Jost function in the case of optical potentials were already studied by [9], for instance, in cases where potentials are more general than CSM ones. In [9] the ABC theorems could not be used and multiple poles were not excluded.…”

“…Locations of zeroes of the Jost function in the case of optical potentials were already studied by [9], for instance, in cases where potentials are more general than CSM ones. In [9] the ABC theorems could not be used and multiple poles were not excluded. Our present CSM case, however, through the argument of concentric circular arcs, clearly gives a transparent and safe solution for the simple nature and the locations of Jost function zeroes.…”

Time evolution of decay for purely absorptive potentials: The effect of spectral singularities

“…Hence, for a given pole at k p = α p − iβ p , with α p and β p positive quantities, there corresponds a pole at k −p = −k * p [18]. Bound and antibound poles correspond, respectively, to purely imaginary positive and negative values of k. However, for complex potentials, time reversal considerations no longer apply [14,19]. For absorptive potentials, causality prevents that poles sit on the first quadrant of the k plane, however, they might appear on the other quadrants.…”