Shnol type theorem for the Agmon ground state - the discrete setting

07.11.2018, 16:15 Uhr  –  Campus Golm, Haus 9, Raum 2.09.1.10
Forschungsseminar Diskrete Spektraltheorie

Siegfried Beckus (Universität Potsdam)

The celebrated Shnol theorem asserts that every polynomially bounded generalized eigenfunction for a given energy E associated with a Schrödinger operator H implies that E is in the L2-spectrum of H. Later B. Simon rediscorvered this result independently and proved additionally that the set of energies admiting a polynomially bounded generalized eigenfunction is dense in the spectrum.

It was conjectured by B. Devyver, M. Fraas, and Y. Pinchover that the polynomial bound on the generalized eigenfunction can be replaced by an object intrinsically defined by H, namely, the Agmon ground state. During
the talk, we positively answer the conjecture. Specically, we show that if u is a generalized eigenfunction
for the eigenvalue E that is bounded by the Agmon ground state then E belongs to the spectrum of H. Throughout the talk, we focus on discrete Schrödinger operators for which a suitable notion of Agmon ground state is available.