A Conditionally Exactly Solvable 1D Dirac Pseudoscalar Interaction Potential

We study an analytically solvable pseudoscalar interaction potential for the one-dimensional stationary Dirac equation, which consists of power terms proportional to \({{x}^{{ - 1}}}\), \({{x}^{{ - 1/3}}}\), and \({{x}^{{1/3}}}\). This potential is classified as conditionally exactly solvable due to the fixed strength of the first term at a specific constant. We present the general solution to the Dirac equation in terms of non-integer index Hermite functions, which are distinct from the conventional integer index Hermite polynomials. We analyze the energy spectrum of the bound states and the eigenfunctions and compare the results with the case without the \({{x}^{{ - 1/3}}}\) term.