On negative eigenvalues of 1D Schrödinger operators with ${\delta }^{\mathrm{\prime }}$-like potentials

In this paper, we investigate negative eigenvalues of exactly solvable quantum models, particularly one-dimensional Hamiltonians with ${\delta }^{\mathrm{\prime }}$-like potentials used to represent localized dipoles. These operators arise as norm resolvent limits of Schrödinger operators with suitably regularized potentials. Although the limiting operator is bounded below, we show that the approximating operators may possess a finite but arbitrarily large number of negative eigenvalues that diverge to $-\mathrm{\infty }$ as the regularization parameter vanishes. This phenomenon illustrates a spectral instability of Schrödinger operators with ${\delta }^{\mathrm{\prime }}$-like singularities.