Perturbation theory for a polyharmonic operator with nonsmooth periodic potential

1991 ◽  
Vol 54 (3) ◽  
pp. 911-915
Author(s):  
Yu. E. Karpeshina
Keyword(s):  
Perturbation Theory ◽  
Periodic Potential ◽  
Polyharmonic Operator

Solution of the Dirac Equation for the Rectilinear Periodic Motion of an Electron

1969 ◽  
Vol 24 (3) ◽  
pp. 344-349
Author(s):  
A. D. Jannussis
Keyword(s):  
Perturbation Theory ◽  
Dirac Equation ◽  
Periodic Motion ◽  
Periodic Potential ◽  
Bragg Reflection ◽  
Wave Number ◽  
Periodic Functions ◽  
Energy Bands ◽  
Energy Eigenvalues ◽  
The Hill

AbstractIn this paper the Dirac equation for a rectilinear onedimensional periodic potential is treated. It is shown that the energy eigenvalues are periodic functions of the wave number Kϰ and the continuous spectrum is split into energy bands. The end points of the energy bands are the points where the Bragg reflection takes place. These results are obtained by perturbation theory, as well as by the method of determinants, since the resulting eigenvalue equation has the form of a determinant which is similar to the Hill determinant.

Kramers' theory for diffusion on a periodic potential

2016 ◽  
Vol 195 ◽  
pp. 111-138 ◽  
Author(s):  
Reuven Ianconescu ◽  
Eli Pollak
Keyword(s):  
Numerical Simulation ◽  
Perturbation Theory ◽  
Diffusion Coefficient ◽  
Normal Mode ◽  
Periodic Potential ◽  
Simulation Data ◽  
Barrier Heights ◽  
Classical Perturbation Theory ◽  
Finite Barrier ◽  
Harmonic Bath

Kramers' turnover theory, based on the dynamics of the collective unstable normal mode (also known as PGH theory), is extended to the motion of a particle on a periodic potential interacting bilinearly with a dissipative harmonic bath. This is achieved by considering the small parameter of the problem to be the deviation of the collective bath mode from its value along the reaction coordinate, defined by the unstable normal mode. With this change, the effective potential along the unstable normal mode remains periodic, albeit with a renormalized mass, or equivalently a renormalized lattice length. Using second order classical perturbation theory, this not only enables the derivation of the hopping rates and the diffusion coefficient, but also the derivation of finite barrier corrections to the theory. The analytical results are tested against numerical simulation data for a simple cosine potential, ohmic friction, and different reduced barrier heights.

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