Abstract We present the exact solution of the 1D Dirac equation for the inverse-square-root potential 1 / x $1/\sqrt{x}$ for several configurations of vector, pseudo-scalar, and scalar fields. Each fundamental solution of the problem can be written as an irreducible linear combination of two Hermite functions of a scaled and shifted argument. We derive the exact equations for bound-state energy eigenvalues and construct accurate approximations for the energy spectrum.

中文翻译：

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反平方根势 1/x$1/\sqrt{x}$ 的一维狄拉克方程的精确解

摘要 我们给出了向量、伪标量和标量场的几种配置的反平方根势 1 / x $1/\sqrt{x}$ 的一维狄拉克方程的精确解。问题的每个基本解都可以写成两个具有缩放和移位参数的 Hermite 函数的不可约线性组合。我们推导出边界态能量特征值的精确方程并构建能量谱的精确近似值。