Quasinormal modes and self-adjoint extensions of the Schrödinger operator

Abstract

We revisit here the analytical continuation approach usually employed to compute quasinormal modes (QNM) and frequencies of a given potential barrier $V$ starting from the bounded states and respective eigenvalues of the Schroedinger operator associated with the potential well corresponding to the inverted potential $-V$. We consider an exactly soluble problem corresponding to a potential barrier of the Poschl-Teller type with a well defined and behaved QNM spectrum, but for which the associated Schroedinger operator $\cal H$ obtained by analytical continuation fails to be self-adjoint. Although $\cal H$ admits self-adjoint extensions, we show that the eigenstates corresponding to the analytically continued QNM do not belong to any self-adjoint extension domain and, consequently, they cannot be interpreted as authentic quantum mechanical bounded states. Our result challenges the practical use of the this type of method when $\cal H$ fails to be self-adjoint since, in such cases, we would not have in advance any reasonable criterion to choose the initial eigenstates of $\cal H$ which would correspond to the analytically continued QNM.