Asymptotic condition in the Lorentz gauge

1961 ◽  
Vol 21 (5) ◽  
pp. 867-868
Author(s):  
M. Wellner
Keyword(s):  
Asymptotic Condition ◽  
Lorentz Gauge

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
Image Position

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

The Lorentz gauge in non-relativistic quantum electrodynamics

1982 ◽  
Vol 383 (1785) ◽  
pp. 485-502 ◽  
Keyword(s):  
Quantum Electrodynamics ◽  
Lamb Shift ◽  
Relativistic Quantum ◽  
Scalar Potential ◽  
Coulomb Energy ◽  
Indefinite Metric ◽  
Longitudinal Field ◽  
Lorentz Gauge ◽  
Transverse And Longitudinal Components ◽  
Relativistic Electrodynamics

It is shown how the conventional Lagrangian of non-relativistic electrodynamics leads to a theory in the Lorentz gauge where the scalar potential is treated on an equal footing with the transverse and longitudinal components of the vector potential. This requires the introduction of an indefinite metric as in the Gupta-Bleuler method. Calculations based on this approach with the use of ordinary perturbation theory for the free-space Lamb-shift of hydrogen are shown to exhibit remarkable exact cancellations between parts of the contribution arising from the scalar field and the entire contribution from the longitudinal field to order e 2 , and the result is in agreement with Bethe’s expression where only transverse photons are involved. The non-relativistic theory in the Lorentz gauge is also used to compute the order- e 2 potential on a charged particle outside a conductor where again similar exact cancellations are exhibited. The advantage of the formalism in the Lorentz gauge is emphasized in that it provides an unambiguous procedure for the evaluation of the leading Coulomb energy shifts particularly in the interaction of particles with the surfaces of active media where the Coulomb gauge may be problematical.

Variational Methods on Finite Dimensional Banach Spaces and Discrete Problems

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Gabriele Bonanno ◽  
Pasquale Candito ◽  
Giuseppina D’Aguí
Keyword(s):  
Variational Methods ◽  
Nonlinear Term ◽  
Asymptotic Condition ◽  
Multiplicity Results ◽  
Order Difference ◽  
Existence And Multiplicity ◽  
Finite Dimensional ◽  
Superlinear Growth ◽  
Discrete Problems ◽  
Minimum Theorem

AbstractIn this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.

Nonuniqueness in theS-matrix reduction by means of the asymptotic condition — I

1964 ◽  
Vol 32 (5) ◽  
pp. 1202-1216
Author(s):  
F. Kaschluhn ◽  
E. Wieczorek
Keyword(s):  
Asymptotic Condition ◽  
Matrix Reduction ◽  
Thes Matrix

scholarly journals A Theorem of Galambos-Bojanić-Seneta Type

2009 ◽  
Vol 2009 ◽  
pp. 1-6 ◽  
Author(s):  
Dragan Djurčić ◽  
Aleksandar Torgašev
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Condition ◽  
The Given

In the theorems of Galambos-Bojanić-Seneta's type, the asymptotic behavior of the functions , for , is investigated by the asymptotic behavior of the given sequence of positive numbers , as and vice versa. The main result of this paper is one theorem of such a type for sequences of positive numbers which satisfy an asymptotic condition of the Karamata type  , for .

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