Page 379 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 379
SPECTRAL THEORY AND INTEGRABLE SYSTEMS (MS-57)
Stahl–Totik regularity for continuum Schrödinger operators
Benjamin Eichinger, benjamin.eichinger@jku.at
Johannes Kepler University, Linz, Austria
Coauthor: Milivoje Lukic´
We develop a theory of Stahl–Totik regularity for half-line Schrödinger operators −∂x2 + V
with bounded potentials (in a local L1 sense). We prove a universal thickness result for the
essential spectrum, E, in the language of potential theory. Namely, E is an Akhiezer-Levin
s√et−aznd+th√ae−EMz +artoi(n√f1−unz )ctaiosnz of the complementary domain at ∞ obeys an asymptotic expansion
→ −∞. The constant aE plays the role of a Robin constant suited
for Schödinger operators. Stahl-Totik regularity is characterized in terms of the behavior of the
x
averages 1 0 V (t)dt and root asymptotics of the Dirichlet solutions as x → ∞. Moreover, it
x
is connected to the zero counting measure for finite truncations. Applications to decaying and
ergodic potentials will be discussed.
On a weighted inequality for fractional integrals
Giorgi Imerlishvili, imerlishvili18@gmail.com
Georgian Technical University, Georgia
Coauthor: Alexander Meskhi
The weighted inequality
Iαf LVq ≤ C f Lp (1)
for the Riesz potential Iα, 0 < α < n, plays an important role in the theory of PDEs. It is worth
mentioning its applications to the theory of Sobolev embeddings (see, e.g., [3]), its connection
with eigenvalue estimates for the Schrödinger operator H = −∆ − V with a potential V (see,
e.g., [1], PP. 91-94), etc. In 1972 D. Adams proved that the above mentioned weighted inequal-
ity holds for 1 < p < q < ∞ if and only if there is a positive constant C such that for all balls
B in Rn,
V (B) ≤ C|B|(1/p−α/n)q.
In the diagonal case, i.e., when p = q, necessity of this condition remains valid, however, it is
not sufficient for (1) (see, e.g., [2]). We proved that the condition
V (B) ≤ C|B|1−pα/n
is simultaneously necessary and sufficient for the boundedness of Iα from the Lorentz space
Lp,1 to the weighted Lebesgue space LVp . Some other related results are also derived.
Acknowledgment. The work was supported by the Shota Rustaveli National Foundation grant
of Georgia (Project No. DI-18-118).
References
[1] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function
spaces, Regional Conference Series in Mathematics, Vol. 79, Amer. Math. Soc. Providence.
Rl, 1991.
377
Stahl–Totik regularity for continuum Schrödinger operators
Benjamin Eichinger, benjamin.eichinger@jku.at
Johannes Kepler University, Linz, Austria
Coauthor: Milivoje Lukic´
We develop a theory of Stahl–Totik regularity for half-line Schrödinger operators −∂x2 + V
with bounded potentials (in a local L1 sense). We prove a universal thickness result for the
essential spectrum, E, in the language of potential theory. Namely, E is an Akhiezer-Levin
s√et−aznd+th√ae−EMz +artoi(n√f1−unz )ctaiosnz of the complementary domain at ∞ obeys an asymptotic expansion
→ −∞. The constant aE plays the role of a Robin constant suited
for Schödinger operators. Stahl-Totik regularity is characterized in terms of the behavior of the
x
averages 1 0 V (t)dt and root asymptotics of the Dirichlet solutions as x → ∞. Moreover, it
x
is connected to the zero counting measure for finite truncations. Applications to decaying and
ergodic potentials will be discussed.
On a weighted inequality for fractional integrals
Giorgi Imerlishvili, imerlishvili18@gmail.com
Georgian Technical University, Georgia
Coauthor: Alexander Meskhi
The weighted inequality
Iαf LVq ≤ C f Lp (1)
for the Riesz potential Iα, 0 < α < n, plays an important role in the theory of PDEs. It is worth
mentioning its applications to the theory of Sobolev embeddings (see, e.g., [3]), its connection
with eigenvalue estimates for the Schrödinger operator H = −∆ − V with a potential V (see,
e.g., [1], PP. 91-94), etc. In 1972 D. Adams proved that the above mentioned weighted inequal-
ity holds for 1 < p < q < ∞ if and only if there is a positive constant C such that for all balls
B in Rn,
V (B) ≤ C|B|(1/p−α/n)q.
In the diagonal case, i.e., when p = q, necessity of this condition remains valid, however, it is
not sufficient for (1) (see, e.g., [2]). We proved that the condition
V (B) ≤ C|B|1−pα/n
is simultaneously necessary and sufficient for the boundedness of Iα from the Lorentz space
Lp,1 to the weighted Lebesgue space LVp . Some other related results are also derived.
Acknowledgment. The work was supported by the Shota Rustaveli National Foundation grant
of Georgia (Project No. DI-18-118).
References
[1] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function
spaces, Regional Conference Series in Mathematics, Vol. 79, Amer. Math. Soc. Providence.
Rl, 1991.
377
