5,513 research outputs found
Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems
We consider solution operators of linear ordinary boundary problems with "too
many" boundary conditions, which are not always solvable. These generalized
Green's operators are a certain kind of generalized inverses of differential
operators. We answer the question when the product of two generalized Green's
operators is again a generalized Green's operator for the product of the
corresponding differential operators and which boundary problem it solves.
Moreover, we show that---provided a factorization of the underlying
differential operator---a generalized boundary problem can be factored into
lower order problems corresponding to a factorization of the respective Green's
operators. We illustrate our results by examples using the Maple package
IntDiffOp, where the presented algorithms are implemented.Comment: 19 page
Special functions from quantum canonical transformations
Quantum canonical transformations are used to derive the integral
representations and Kummer solutions of the confluent hypergeometric and
hypergeometric equations. Integral representations of the solutions of the
non-periodic three body Toda equation are also found. The derivation of these
representations motivate the form of a two-dimensional generalized
hypergeometric equation which contains the non-periodic Toda equation as a
special case and whose solutions may be obtained by quantum canonical
transformation.Comment: LaTeX, 24 pp., Imperial-TP-93-94-5 (revision: two sections added on
the three-body Toda problem and a two-dimensional generalization of the
hypergeometric equation
Electromagnetic Potential in Pre-Metric Electrodynamics: Causal Structure, Propagators and Quantization
An axiomatic approach to electrodynamics reveals that Maxwell electrodynamics
is just one instance of a variety of theories for which the name
electrodynamics is justified. They all have in common that their fundamental
input are Maxwell's equations (or ) and
and a constitutive law H = # F which relates the field
strength two-form and the excitation two-form . A local and linear
constitutive law defines what is called local and linear pre-metric
electrodynamics whose best known application are the effective description of
electrodynamics inside media including, e.g., birefringence. We analyze the
classical theory of the electromagnetic potential before we use methods
familiar from mathematical quantum field theory in curved spacetimes to
quantize it in a locally covariant way. Our analysis of the classical theory
contains the derivation of retarded and advanced propagators, the analysis of
the causal structure on the basis of the constitutive law (instead of a metric)
and a discussion of the classical phase space. This classical analysis sets the
stage for the construction of the quantum field algebra and quantum states.
Here one sees, among other things, that a microlocal spectrum condition can be
formulated in this more general setting.Comment: 34 pages, references added, update to published version, title
updated to published versio
Potentials for -quasiconvexity
We show that each constant rank operator admits an exact
potential in frequency space. We use this fact to show that the
notion of -quasiconvexity can be tested against compactly
supported fields. We also show that -free Young measures are
generated by sequences , modulo shifts by the barycentre.Comment: 15 pages; to appear in Calculus of Variations and Partial
Differential Equation
- …