Antibound States and Exponentially Decaying Sturm–Liouville Potentials

We consider the Sturm–Liouville equation y ″( x )+{λ− q ( x )} y ( x )=0 (0⩽ x <∞) (1.1) with a boundary condition at x = 0 which can be either the Dirichlet condition y (0)=0 (1.2) or the Neumann condition y ′(0)=0 (1.3) As usual, λ is the complex spectral parameter with 0 ⩽ arg λ < 2π, and the potential q is real‐valued and locally integrable in [0, ∞).