Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials

We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie on the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-side estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also discuss convergence rates of the method as well as illustrate our results with various numerical experiments.

[1]  Lyonell Boulton,et al.  On approximation of the eigenvalues of perturbed periodic Schrödinger operators , 2007, math/0702420.

[2]  Michael Levitin,et al.  Spectral pollution and second-order relative spectra for self-adjoint operators , 2002 .

[3]  Mathieu Lewin,et al.  Variational methods in relativistic quantum mechanics , 2007, 0706.3309.

[4]  Eugene Shargorodsky,et al.  Geometry of higher order relative spectra and projection methods , 2000 .

[5]  P. Hislop,et al.  Exponential decay of two-body eigenfunctions: a review. , 2000 .

[6]  K. Dyall,et al.  Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set , 1990 .

[7]  A. Berthier,et al.  On the point spectrum of Dirac operators , 1987 .

[8]  Hardy inequalities in function spaces , 1999 .

[9]  Jean Dolbeault,et al.  On the eigenvalues of operators with gaps. Application to Dirac operators. , 2000, 2206.06327.

[10]  Jean Dolbeault,et al.  A variational method for relativistic computations in atomic and molecular physics , 2003 .

[11]  H. Langer,et al.  Variational principles for eigenvalues of block operator matrices , 2002 .

[12]  Bernd Thaller,et al.  The Dirac Equation , 1992 .

[13]  Richard E. Stanton,et al.  Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations , 1984 .

[14]  J. Dolbeault,et al.  Relativistic Hydrogenic Atoms in Strong Magnetic Fields , 2006, math/0607027.

[15]  Lyonell Boulton.  NON-VARIATIONAL APPROXIMATION OF DISCRETE EIGENVALUES OF SELF-ADJOINT OPERATORS , 2005 .

[16]  J. Dolbeault,et al.  Minimization methods for the one-particle dirac equation. , 2000, Physical review letters.

[17]  Lyonell Boulton.  Limiting set of second order spectra , 2006, Math. Comput..

[18]  M. Griesemer,et al.  ACCUMULATION OF DISCRETE EIGENVALUES OF THE RADIAL DIRAC OPERATOR , 1999 .

[19]  Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit , 2001, math/0111115.