Antibound states for a class of one-dimensional Schrödinger operators

Let δ(x) be the Dirac's delta,q(x)∈L1(R)∩L2(R) be a real valued function, and λ, μ∈R; we will consider the following class of one-dimensional formal Schrodinger operators on\(L{^2} (R)\tilde H(\lambda ,\mu ) = - (d{^2} /dx{^2} ) + \lambda \delta (x) + \mu q(x)\). It is known that to the formal operator\(\tilde H(\lambda ,\mu )\) may be associated a selfadjoint operatorH(λ,μ) onL2(R). Ifq is of finite range, for λ>0 and |μ| is small enough, we prove thatH(λ,μ) has an antibound state; that is the resolvent ofH(λ,μ) has a pole on the negative real axis on the second Riemann sheet.