Abstract
We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(R). The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Heisenberg–Weyl group and some of their extensions.
Highlights
We study the relations between certain physical relevant lowdimensional Lie groups, in connection to affine transformations on the whole real line (R) and their representations on the Hilbert space L2 (R) as well as to other notions as the Hermite functions, other bases in L2 (R) and the eigenfunctions of the Fourier transform
As a consequence of these relations, some invariance properties are disclosed. These invariance properties come from the options between four types of freedom. These are: (i) the freedom to choose between coordinate and momentum representations and the respective bases determined by each of these representations; (ii) the freedom to choose an origin on the real line when using any of these two representations; (iii) the freedom to choose the units of length on R; and (iv) the freedom to choose an orientation on the line
We considered the symmetry under Fourier Transform for the Hermite functions
Summary
We study the relations between certain physical relevant lowdimensional Lie groups, in connection to affine transformations on the whole real line (R) and their representations on the Hilbert space L2 (R) as well as to other notions as the Hermite functions, other bases in L2 (R) and the eigenfunctions of the Fourier transform. The conclusion is that X and P, along with the central operator I, determine the Lie algebra for the Heisenberg–Weyl group H (1) In this context, we say that the real line, meaning the space L2 (R), supports a unitary representation of H (1). Consistency with Fourier transform invariance implies that k0 = k−1 This suggests a result that shall become evident soon, which is that the algebra describing e o (1), i.e., the Heisenberg–Weyl group enlarged the invariance in the real line has to be H with dilatations. We need to introduce orientation invariance and negative values of k for dilatations in our picture This is done by means of the parity operator P , where the action of P on the continuous bases is given by P | x i = | − x i and P | pi = | − pi. We give our concluding remarks in the final Section 6
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