On Some Linear and Nonlinear Eigenvalue Problems in Relativistic Quantum Chemistry

Abstract

In relativistic quantum mechanics [1], the bound states of an electron under the action of an external electrostatic potential V are represented by the wave functions φ ∈ L ² (ℝ³, ℂ⁴) which are solutions of the equation

$$ (H_0 + V)\varphi = \lambda \varphi ,\lambda \in \mathbb{R} $$

((1.1))

$$ with H_0 = - ich\sum\limits_{k = 1}^3 {\alpha _k \partial _k + mc^2 \beta ,} $$

((1.2))

where c denotes the speed of light, m > 0, the mass of the electron, and h is Planck’s constant. Moreover, α₁, α₂, α₃and β are 4 × 4 complex matrices, whose standard form (in 2 × 2 blocks) is

$$ \beta = \left( {\begin{array}{*{20}c} I & 0 \\ 0 & { - I} \\ \end{array} } \right), \alpha _k = \left( {\begin{array}{*{20}c} 0 & {\sigma _k } \\ {\sigma _k } & 0 \\ \end{array} } \right) (k = 1,2,3), $$

with

$$ \sigma _1 = \left( {\begin{array}{*{20}c} 0 & 1 \\ 1 & 0 \\ \end{array} } \right), \sigma _2 = \left( {\begin{array}{*{20}c} 0 & { - i} \\ i & 0 \\ \end{array} } \right),\sigma _3 = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & { - 1} \\ \end{array} } \right). $$

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