arXiv subjects
Publications and source records attributed to Charles Schwartz.
Recent innovations on the differential calculus for functions of non-commuting variables, begun for a quaternionic variable, are now extended to the case of a general matrix over the complex numbers. The expansion of F(X+Delta) is given to first order in Delta for general matrix variables X and Delta that do not commute with each other.
Here we follow the basic analysis that is common for real and complex variables and find how it can be applied to a quaternionic variable. Non-commutativity of the quaternion algebra poses obstacles for the usual manipulations; but we show how many of those obstacles can be overcome. After a tiny bit of linear algebra we look at the beginnings of differential calculus. The surprising result is that the first order term in the expansion of F(x+delta) is a compact formula involving both F'(x) and [F(x) - F(x*)]/(x-x*).
Further results are reported for the one-component quaternionic wave equation recently introduced. A Lagrangian is found for the momentum-space version of the free equation; and another, nonlocal in time, is found for the complete equation. Further study of multi-particle systems has us looking into the mathematics of tensor products of Hilbert spaces. The principles of linearity and superposition are also clarified to good effect in advancing the quaternionic theory.
We study a one-component quaternionic wave equation which is relativistically covariant. Bi-linear forms include a conserved 4-current and an antisymmetric second rank tensor. Waves propagate within the light-cone and there is a conserved quantity which looks like helicity. The principle of superposition is retained in a slightly altered manner. External potentials can be introduced in a way that allows for gauge invariance. There are some results for scattering theory and for two-particle wavefunctions as well as the beginnings of second quantization. However, we are unable to find a suitable Lagrangian or an energy-momentum tensor.
Recently reported computations have been extended to give ten more decimals of accuracy in the ground state energy of the Schrodinger equation for the idealized Helium atom. With the F basis - Hylleraas coordinates with negative powers and a logarithm of s - carried to the fiftieth order (N = 24,099 terms) we find the eigenvalue E = -2.90372 43770 34119 59831 11592 45194 40444 66969 25309 ...
Extensive variational computations are reported for the ground state energy of the non-relativistic two-electron atom. Several different sets of basis functions were systematically explored, starting with the original scheme of Hylleraas. The most rapid convergence is found with a combination of negative powers and a logarithm of the coordinate s = r_{1}+ r_{2}. At N=3091 terms we pass the previous best calculation (Korobov's 25 decimal accuracy with N=5200 terms) and we stop at N=10257 with E = -2.90372 43770 34119 59831 11592 45194 40444 ... Previous mathematical analysis sought to link the convergence rate of such calculations to specific analytic properties of the functions involved. The application of that theory to this new experimental data leaves a rather frustrating situation, where we seem able to do little more than invoke vague concepts, such as ``flexibility.'' We conclude that theoretical understanding here lags well behind the power of available computing machinery.
The algebraic formulation of Large N matrix mechanics recently developed by Halpern and Schwartz leads to a practical method of numerical computation for both action and Hamiltonian problems. The new technique posits a boundary condition on the planar connected parts X_w, namely that they should decrease rapidly with increasing order. This leads to algebraic/variational schemes of computation which show remarkably rapid convergence in numerical tests on some many- matrix models. The method allows the calculation of all moments of the ground state, in a sequence of approximations, and excited states can be determined as well. There are two unexpected findings: a large d expansion and a new selection rule for certain types of interaction.