3,567 research outputs found
A categorical framework for the quantum harmonic oscillator
This paper describes how the structure of the state space of the quantum
harmonic oscillator can be described by an adjunction of categories, that
encodes the raising and lowering operators into a commutative comonoid. The
formulation is an entirely general one in which Hilbert spaces play no special
role. Generalised coherent states arise through the hom-set isomorphisms
defining the adjunction, and we prove that they are eigenstates of the lowering
operators. Surprisingly, generalised exponentials also emerge naturally in this
setting, and we demonstrate that coherent states are produced by the
exponential of a raising morphism acting on the zero-particle state. Finally,
we examine all of these constructions in a suitable category of Hilbert spaces,
and find that they reproduce the conventional mathematical structures.Comment: 44 pages, many figure
Fourier duality for fractal measures with affine scales
For a family of fractal measures, we find an explicit Fourier duality. The
measures in the pair have compact support in \br^d, and they both have the
same matrix scaling. But the two use different translation vectors, one by a
subset in \br^d, and the other by a related subset . Among other
things, we show that there is then a pair of infinite discrete sets
and in \br^d such that the -Fourier exponentials are
orthogonal in , and the -Fourier exponentials are
orthogonal in . These sets of orthogonal "frequencies" are
typically lacunary, and they will be obtained by scaling in the large. The
nature of our duality is explored below both in higher dimensions and for
examples on the real line.
Our duality pairs do not always yield orthonormal Fourier bases in the
respective -Hilbert spaces, but depending on the geometry of certain
finite orbits, we show that they do in some cases. We further show that there
are new and surprising scaling symmetries of relevance for the ergodic theory
of these affine fractal measures.Comment: v
The reproducing kernel structure arising from a combination of continuous and discrete orthogonal polynomials into Fourier systems
We study mapping properties of operators with kernels defined via a
combination of continuous and discrete orthogonal polynomials, which provide an
abstract formulation of quantum (q-) Fourier type systems. We prove Ismail
conjecture regarding the existence of a reproducing kernel structure behind
these kernels, by establishing a link with Saitoh theory of linear
transformations in Hilbert space. The results are illustrated with Fourier
kernels with ultraspherical weights, their continuous q-extensions and
generalizations. As a byproduct of this approach, a new class of sampling
theorems is obtained, as well as Neumann type expansions in Bessel and q-Bessel
functions.Comment: 16 pages; Title changed, major reformulations. To appear in Constr.
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