15,268 research outputs found
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]
Quantum Dot in Z-shaped Graphene Nanoribbon
Stimulated by recent advances in isolating graphene, we discovered that
quantum dot can be trapped in Z-shaped graphene nanoribbon junciton. The
topological structure of the junction can confine electronic states completely.
By varying junction length, we can alter the spatial confinement and the number
of discrete levels within the junction. In addition, quantum dot can be
realized regardless of substrate induced static disorder or irregular edges of
the junction. This device can be used to easily design quantum dot devices.
This platform can also be used to design zero-dimensional functional nanoscale
electronic devices using graphene ribbons.Comment: 4 pages, 3 figure
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]
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