635 research outputs found
A Class of Exactly-Solvable Eigenvalue Problems
The class of differential-equation eigenvalue problems
() on the interval
can be solved in closed form for all the eigenvalues and
the corresponding eigenfunctions . The eigenvalues are all integers and
the eigenfunctions are all confluent hypergeometric functions. The
eigenfunctions can be rewritten as products of polynomials and functions that
decay exponentially as . For odd the polynomials that are
obtained in this way are new and interesting classes of orthogonal polynomials.
For example, when N=1, the eigenfunctions are orthogonal polynomials in
multiplying Airy functions of . The properties of the polynomials for all
are described in detail.Comment: REVTeX, 16 pages, no figur
Ladder Operators for Quantum Systems Confined by Dihedral Angles
We report the identification and construction of raising and lowering
operators for the complete eigenfunctions of isotropic harmonic oscillators
confined by dihedral angles, in circular cylindrical and spherical coordinates;
as well as for the hydrogen atom in the same situation of confinement, in
spherical, parabolic and prolate spheroidal coordinates. The actions of such
operators on any eigenfunction are examined in the respective coordinates,
illustrating the possibility of generating the complete bases of eigenfunctions
in the respective coordinates for both physical systems. The relationships
between the eigenfunctions in each pair of coordinates, and with the same
eigenenergies are also illustrated
Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation
The relation between the Wilson-Polchinski and the Litim optimized ERGEs in
the local potential approximation is studied with high accuracy using two
different analytical approaches based on a field expansion: a recently proposed
genuine analytical approximation scheme to two-point boundary value problems of
ordinary differential equations, and a new one based on approximating the
solution by generalized hypergeometric functions. A comparison with the
numerical results obtained with the shooting method is made. A similar accuracy
is reached in each case. Both two methods appear to be more efficient than the
usual field expansions frequently used in the current studies of ERGEs (in
particular for the Wilson-Polchinski case in the study of which they fail).Comment: Final version to appear in Nucl. Phys. B. Some references added
correctl
A dimensionally continued Poisson summation formula
We generalize the standard Poisson summation formula for lattices so that it
operates on the level of theta series, allowing us to introduce noninteger
dimension parameters (using the dimensionally continued Fourier transform).
When combined with one of the proofs of the Jacobi imaginary transformation of
theta functions that does not use the Poisson summation formula, our proof of
this generalized Poisson summation formula also provides a new proof of the
standard Poisson summation formula for dimensions greater than 2 (with
appropriate hypotheses on the function being summed). In general, our methods
work to establish the (Voronoi) summation formulae associated with functions
satisfying (modular) transformations of the Jacobi imaginary type by means of a
density argument (as opposed to the usual Mellin transform approach). In
particular, we construct a family of generalized theta series from Jacobi theta
functions from which these summation formulae can be obtained. This family
contains several families of modular forms, but is significantly more general
than any of them. Our result also relaxes several of the hypotheses in the
standard statements of these summation formulae. The density result we prove
for Gaussians in the Schwartz space may be of independent interest.Comment: 12 pages, version accepted by JFAA, with various additions and
improvement
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