Relativistic Comparison Theorems
arXiv:1004.1109 · doi:10.1103/PhysRevA.81.052101
Abstract
Comparison theorems are established for the Dirac and Klein--Gordon equations. We suppose that V^{(1)}(r) and V^{(2)}(r) are two real attractive central potentials in d dimensions that support discrete Dirac eigenvalues E^{(1)}_{k_dν} and E^{(2)}_{k_dν}. We prove that if V^{(1)}(r) \leq V^{(2)}(r), then each of the corresponding discrete eigenvalue pairs is ordered E^{(1)}_{k_dν} \leq E^{(2)}_{k_dν}. This result generalizes an earlier more restrictive theorem that required the wave functions to be node free. For the the Klein--Gordon equation, similar reasoning also leads to a comparison theorem provided in this case that the potentials are negative and the eigenvalues are positive.
6 pages
References in corpus (2)
Cited by in corpus (11)
- Dirac Equation with Spin Symmetry for the Modified Pöschl-Teller Potential in -dimensions
- A general comparison theorem
- The auxiliary field method in quantum mechanics
- Comparison theorems for the Dirac equation with spin-symmetric and pseudo-spin-symmetric interactions
- Refined comparison theorems for the Dirac equation in d dimensions
- Dirac eigenvalues for a softcore Coulomb potential in d dimensions
- Nodal theorems for the Dirac equation in d >= 1 dimensions
- Geometric spectral inversion for singular potentials
- Comparison theorems for the position-dependent mass Schroedinger equation
- General comparison theorems for the Klein-Gordon equation in d dimensions
- Sharp comparison theorems for the Klein--Gordon equation in dimensions