The maximal 'kinematical' invariance group for an arbitrary potential revised

Group classification of one particle Schr\"odinger equations with arbitrary potentials (C. P. Boyer, Helv. Phys. Acta {\bf 47}, p. 450, 1974) is revised. The corrected completed list of non-equivalent potentials and the corresponding symmetries is presented together with exact identification of symmetry algebras and admissible equivalence transformations.Comment: Misprints are correcte

Traces on Infinite-Dimensional Brauer Algebras

We describe the central measures for the random walk on graded graphs. Using this description, we obtain the list of all finite traces on three infinite-dimensional algebras: on the Brauer algebra, on the partition algebra, and on the walled Brauer algebra. For the first two algebras, these lists coincide with the list of all finite traces of the infinite symmetric group. For the walled Brauer algebra, the list of finite traces coincide with the list of finite traces of the square of the latter group. We introduce the operation which corresponds to the graph another graph which called "Pasclization" of the initial graph and then give the general criteria for coinsidness of the sets of traces on both graphs.Comment: 9 pages, 20 Re

Integrability and supersymmetry of Schroedinger-Pauli equations for neutral particles

Integrable quantum mechanical systems for neutral particles with spin $\frac12$ and nontrivial dipole momentum are classified. It is demonstrated that such systems give rise to new exactly solvable problems of quantum mechanics with clear physical content. Solutions for three of them are given in explicit form. The related symmetry algebras and superalgebras are discussed. The presented classification is restricted to two-dimensional systems which admit matrix integrals of motion linear in momenta.Comment: 22 page

An interpretation of the infrared singularity of the effective electromagnetic field

The problem of infrared divergence of the effective electromagnetic field produced by elementary particles is revisited using the non-equilibrium model of an electron interacting with low-temperature photons. It is argued that the infrared singularity of the effective field can be interpreted as a thermalization of the electron. It is shown that this thermalization is negligible in actual field measurements as it is completely dominated by the usual quantum spreading.Comment: 8 pages, 3 figure