Quasinormal modes and self-adjoint extensions of the Schroedinger operator

• URL: http://arxiv.org/abs/2010.10674v2
• Date: Mon, 11 Jan 2021 18:03:37 GMT
• Title: Quasinormal modes and self-adjoint extensions of the Schroedinger operator
• Authors: J\'ulio C. Fabris, Mart\'in G. Richarte, Alberto Saa
• Abstract summary: We revisit the analytical continuation approach usually employed to compute quasinormal modes. We show that the eigenstates corresponding to the analytically continued QNM do not belong to any self-adjoint extension domain.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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