Maximal subalgebras of vector fields for equivariant quantizations

in Journal of Mathematical Physics (2001), 42(2), 582-589

The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these ... [more ▼]

The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables that are polynomial in the momenta and differential operators. Here, we determine which finite dimensional graded Lie subalgebras are maximal. In order to characterize these, we make use of results of Guillemin, Singer, and Sternberg and Kobayashi and Nagano. [less ▲]

Upper bounds for the many-channel Marchenko transformation operator and its derivatives

in Journal of Mathematical Physics (1979), 20

We specify the class of perturbative complex matrix potentials for which the corresponding many-channel Marchenko type transformation operators are bounded and integrable. Our reference matrix potential ... [more ▼]

We specify the class of perturbative complex matrix potentials for which the corresponding many-channel Marchenko type transformation operators are bounded and integrable. Our reference matrix potential contains Coulomb interactions, different threshold energies, and centrifugal potentials with different angular momenta. Estimates for the transformation operator and its derivatives are obtained; they enable us to improve our recent results and are necessary for the establishment of a unique solution to the ''generalized Marchenko fundamental equation.'' From the existence of an integrable transformation operator, the analyticity of the Jost solution as a function of k1 is deduced in the upper-half of the physical k1 plane. [less ▲]

The translation kernel in the n-dimensional scattering problem

in Journal of Mathematical Physics (1977), 18

Radial wavefunctions are defined for the n-dimensional scattering problem (n>~1) with spherical symmetry by conditions of regularity at the origin or by conditions of behavior at infinity. The existence ... [more ▼]

Radial wavefunctions are defined for the n-dimensional scattering problem (n>~1) with spherical symmetry by conditions of regularity at the origin or by conditions of behavior at infinity. The existence of translation kernels can therefore be discussed in both instances. The problem of representing regular solutions appears to be essentially different from that of representing irregular solutions. The essential difference originates from the type of domain used in the representation: It is bounded in the first case and unbounded in the second. If one can still compare the ranges of validity of the two types of representation when one is dealing with a scalar situation, upon proceeding to a matrix situation, a comparison is no longer possible. [less ▲]

The translation kernel in the $n$-dimensional scattering problem

in Journal of Mathematical Physics (1977), 18(11), 2223--2231