Event № 669
The celebrated Shnol theorem  asserts that every polynomially bounded generalized eigenfunction
for a given energy E 2 R associated with a Schrodinger operator H implies that E is
in the L2-spectrum of H. Later Simon  rediscorvered this result independently and proved
additionally that the set of energies admiting 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  that the polynomial bound on the generalized eigenfunction can be
replaced by an object intrinsically dened 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. Specically, we show that if u is a generalized eigenfunction
for the eigenvalue E 2 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.