Graduate Seminar Analysis

Tuesday, July 18, 2017, 2 p.m.

Shnol type Theorem for the Agmon ground state

Siegfried Beckus, Technion-Israel Institute of Technology: t.b.a

Abstract: 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 Simon rediscovered this result independently and proved additionally that the set of energies admitting a polynomially bounded generalized eigenfunction is dense in the spectrum. A remarkable extension of these results holds also in
the Dirichlet setting. It has been conjectured that the polynomial bound on the generalized eigenfunction can be replaced by an object intrinsically defined by H, namely, the Agmon ground state. During the talk, we positively answer the conjecture indicating that the Agmon ground state describes the spectrum of the operator H. Specifically, 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 L2-spectrum of H. Furthermore, this assertion extends to the Dirichlet setting whenever a suitable notion of Agmon ground state is available.

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Ort: Raum 001, Pontdriesch 14-16 , 52062 Aachen