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Register a new userSymmetries of the Free Schrodinger Equation in the Non-Commutative Plane
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We study all the symmetries of the free Schrodinger equation in the non-commutative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schrodinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.
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We revise and extend the algorithm provided in [1] to compute the finite Connes distance between normal states. The original formula in [1] contains an error and actually only provides a lower bound. The correct expression, which we provide here, involves the computation of the infimum of an expression which involves the transverse component of the algebra element in addition to the longitudinal component of [1]. This renders the formula less user-friendly, as the determination of the exact transverse component for which the infimum is reached remains a non-trivial task, but under rather generic conditions it turns out that the Connes distance is proportional to the trace norm of the difference in the density matrices, leading to considerable simplification. In addition, we can determine an upper bound of the distance by emulating and adapting the approach of [2] in our Hilbert-Schmidt operatorial formulation. We then look for an optimal element for which the upper bound is reached. We are able to find one for the Moyal plane through the limit of a sequence obtained by finite dimensional projections of the representative of an element belonging to a multiplier algebra, onto the subspaces of the total Hilbert space, occurring in the spectral triple and spanned by the eigen-spinors of the respective Dirac operator. This is in contrast with the fuzzy sphere, where the upper bound, which is given by the geodesic of a commutative sphere is never reached for any finite $n$-representation of $SU(2)$. Indeed, for the case of maximal non-commutativity ($n = 1/2$), the finite distance is shown to coincide exactly with the above mentioned lower bound, with the transverse component playing no role. This, however starts changing from $n=1$ onwards and we try to improve the estimate of the finite distance and provide an almost exact result, using our new and modified algorithm.
New exact analytical bound-state solutions of the radial Dirac equation in 3+1 dimensions for two sets of couplings and radial potential functions are obtained via mapping onto the nonrelativistic bound-state solutions of the one-dimensional generalized Morse potential. The eigenfunctions are expressed in terms of generalized Laguerre polynomials, and the eigenenergies are expressed in terms of solutions of equations that can be transformed into polynomial equations. Several analytical results found in the literature, including the Dirac oscillator, are obtained as particular cases of this unified approach.
The three-dimensional Schrodinger equation with a position-dependent (effective) mass is studied in the framework of Supersymmetrical (SUSY) Quantum Mechanics. The general solution of SUSY intertwining relations with first order supercharges is obtained without any preliminary constraints. Several forms of coefficient functions of the supercharges are investigated and analytical expressions for the mass function and partner potentials are found. As usual for SUSY Quantum Mechanics with nonsingular superpotentials, the spectra of intertwined Hamiltonians coincide up to zero modes of supercharges, and the corresponding wave functions are connected by intertwining relations. All models are partially integrable by construction: each of them has at least one second order symmetry operator.
In this article, we study the decay of the solutions of Schrodinger equations in the exterior of an obstacle. The main situations we are interested in are the general case (no non-trapping assumptions) or some weakly trapping situations
We construct a new example of the spinning-particle model without Grassmann variables. The spin degrees of freedom are described on the base of an inner anti-de Sitter space. This produces both $Gamma^mu$ and $Gamma^{mu u}$,-matrices in the course of quantization. Canonical quantization of the model implies the Dirac equation. We present the detailed analysis of both the Lagrangian and the Hamiltonian formulations of the model and obtain the general solution to the classical equations of motion. Comparing {it Zitterbewegung} of the spatial coordinate with the evolution of spin, we ask on the possibility of space-time interpretation for the inner space of spin. We enumerate similarities between our analogous model of the Dirac equation and the two-body system subject to confining potential which admits only the elliptic orbits of the order of de Broglie wave-length. The Dirac equation dictates the perpendicularity of the elliptic orbits to the direction of center-of-mass motion.
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