7 research outputs found
Levinson's Theorem for Dirac Particles
Levinson's theorem for Dirac particles constraints the sum of the phase
shifts at threshold by the total number of bound states of the Dirac equation.
Recently, a stronger version of Levinson's theorem has been proven in which the
value of the positive- and negative-energy phase shifts are separately
constrained by the number of bound states of an appropriate set of
Schr\"odinger-like equations. In this work we elaborate on these ideas and show
that the stronger form of Levinson's theorem relates the individual phase
shifts directly to the number of bound states of the Dirac equation having an
even or odd number of nodes. We use a mean-field approximation to Walecka's
scalar-vector model to illustrate this stronger form of Levinson's theorem. We
show that the assignment of bound states to a particular phase shift should be
done, not on the basis of the sign of the bound-state energy, but rather, in
terms of the nodal structure (even/odd number of nodes) of the bound state.Comment: Latex with Revtex, 7 postscript figures (available from the author),
SCRI-06109
A Solution of the Maxwell-Dirac Equations in 3+1 Dimensions
We investigate a class of localized, stationary, particular numerical
solutions to the Maxwell-Dirac system of classical nonlinear field equations.
The solutions are discrete energy eigenstates bound predominantly by the
self-produced electric field.Comment: 12 pages, revtex, 2 figure
Levinson's theorem for the Schr\"{o}dinger equation in two dimensions
Levinson's theorem for the Schr\"{o}dinger equation with a cylindrically
symmetric potential in two dimensions is re-established by the Sturm-Liouville
theorem. The critical case, where the Schr\"{o}dinger equation has a finite
zero-energy solution, is analyzed in detail. It is shown that, in comparison
with Levinson's theorem in non-critical case, the half bound state for
wave, in which the wave function for the zero-energy solution does not decay
fast enough at infinity to be square integrable, will cause the phase shift of
wave at zero energy to increase an additional .Comment: Latex 11 pages, no figure and accepted by P.R.A (in August); Email:
[email protected], [email protected]
The Relativistic Levinson Theorem in Two Dimensions
In the light of the generalized Sturm-Liouville theorem, the Levinson theorem
for the Dirac equation in two dimensions is established as a relation between
the total number of the bound states and the sum of the phase shifts
of the scattering states with the angular momentum :
\noindent The critical case, where the Dirac equation has a finite
zero-momentum solution, is analyzed in detail. A zero-momentum solution is
called a half bound state if its wave function is finite but does not decay
fast enough at infinity to be square integrable.Comment: Latex 14 pages, no figure, submitted to Phys.Rev.A; Email:
[email protected], [email protected]
Levinson's Theorem for the Klein-Gordon Equation in Two Dimensions
The two-dimensional Levinson theorem for the Klein-Gordon equation with a
cylindrically symmetric potential is established. It is shown that
, where denotes
the difference between the number of bound states of the particle
and the ones of antiparticle with a fixed angular momentum , and
the is named phase shifts. The constants and
are introduced to symbol the critical cases where the half bound
states occur at .Comment: Revtex file 14 pages, submitted to Phys. Rev.
Levinson's theorem and scattering phase shift contributions to the partition function of interacting gases in two dimensions
We consider scattering state contributions to the partition function of a
two-dimensional (2D) plasma in addition to the bound-state sum. A partition
function continuity requirement is used to provide a statistical mechanical
heuristic proof of Levinson's theorem in two dimensions. We show that a proper
account of scattering eliminates singularities in thermodynamic properties of
the nonideal 2D gas caused by the emergence of additional bound states as the
strength of an attractive potential is increased. The bound-state contribution
to the partition function of the 2D gas, with a weak short-range attraction
between its particles, is found to vanish logarithmically as the binding energy
decreases. A consistent treatment of bound and scattering states in a screened
Coulomb potential allowed us to calculate the quantum-mechanical second virial
coefficient of the dilute 2D electron-hole plasma and to establish the
difference between the nearly ideal electron-hole gas in GaAs and the strongly
correlated exciton/free-carrier plasma in wide-gap semiconductors such as ZnSe
or GaN.Comment: 10 pages, 3 figures; new version corrects some minor typo