The celebrated Shnol theorem 4
asserts that every polynomially bounded generalized eigenfunction for a given energy E∈R associated with a Schrödinger operator H implies that E is in the L2-spectrum of H. Later Simon 5
rediscorvered 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 hold also in the Dirichlet setting 1, 2
It was conjectured in 3
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∈R 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.
A. Boutet de Monvel, D. Lenz, and P. Stollmann, Sch'nol's theorem for strongly local forms, 189-211, Israel J. Math., vol. 173, 2009.
A. Boutet de Monvel and P. Stollmann, Eigenfunction expansions for generators of Dirichlet forms, 131-144, J. Reine Angew. Math. vol. 561, 2003.
B. Devyver, M. Fraas, and Y. Pinchover, Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon, 4422-4489, J. Funct. Anal., vol. 266, no. 7, 2014.
È. È. Šhnol', On the behavior of the eigenfunctions of Schrödinger's equation, 273-286, Mat. Sb. (N.S.), vol. 42, (84), 1957; erratum, 259, vol. 46, (88), 1957 (Russian).
B. Simon, Spectrum and continuum eigenfunctions of Schrödinger operators, 347-355, J. Funct. Anal., vol. 42, no. 3, 1981.