Dear Colleagues,
On 9 april 2024 starting from 14:00 in room 34, 4th floor of the Department
of Statistical Sciences, Sapienza University of Rome, (building CU002), two
seminars on non-local operators will take place. Here is the program.
Please feel free to inform interested people.
Best regards,
Enrico Scalas
--
14:00 - 15:00 Atsuhide Ishida (Tokyo University of Science, Tokyo, Japan)
Mourre inequality for non-local Schrödinger operators
We consider the Mourre inequality for the following self-adjoint operator
$H=\Psi(-\Delta/2)+V$ acting on $L^2(\mathbb{R}^d)$, where $\Psi:
[0,\infty)\rightarrow\mathbb{R}$ is an increasing function, $\Delta$ is the
Laplacian and $V: \mathbb{R}^d\rightarrow\mathbb{R}$ is an interaction
potential. Mourre inequality immediately yields the discreteness and finite
multiplicity of the eigenvalues. Moreover, the Mourre inequality together
with the limiting absorption principle can be used to show absence of the
singular continuous spectrum. In addition, Mourre inequality is also used
for the proof of the minimal velocity estimate that plays an important role
in scattering theory.
In this talk, we report that Mourre inequality holds under a general $\Psi$
and $V$ by choosing the conjugate operator $A=(p \cdot x + x \cdot p)/2$
with $p= - i \nabla$, and that the discreteness and finite multiplicity of
the eigenvalues hold. This talk is a joint work with J. Lőrinczi (Alfred
Rényi Institute) and I. Sasaki (Shinshu University).
15:00 - 15:30 Coffee break
15:30 - 16:30 József Lőrinczi (Alfred Rényi Institute, Budapest, Hungary)
Embedded eigenvalues for a class of non-local Schrödinger operators
Generally, the spectrum of a non-local Schrödinger operator may be rather
intricate, even when they are self-adjoint operators. In this talk I plan
to discuss some explicit cases when positive or zero eigenvalues occur, and
also address the problem more generally, aiming to describe potentials
which can give rise to zero eigenvalues for massive or massless
(fractional) relativistic Schrodinger operators. If time permits, I intend
to explain how random processes with jumps can be used to analyse such
properties.
16:30 - 18:00 Discussion time