TITLE:
Dispersion Relations in Diffraction in Time
AUTHORS:
Salvador Godoy, Karen Villa
KEYWORDS:
Diffraction in Time, Dispersion Relations, Hilbert Transforms
JOURNAL NAME:
Applied Mathematics,
Vol.15 No.7,
July
26,
2024
ABSTRACT: In agreement with Titchmarsh’s theorem, we prove that dispersion relations are just the Fourier-transform of the identity,
g(
x
′
)=±Sgn(
x
′
)g(
x
′
)
, which defines the property of being a truncated functions at the origin. On the other hand, we prove that the wave-function of a generalized diffraction in time problem is just the Fourier-transform of a truncated function. Consequently, the existence of dispersion relations for the diffraction in time wave-function follows. We derive these explicit dispersion relations.