Himeji Conference on Partial Differential Equations
Titles and abstracts 2022
KAIZUKA, Koichi (Nippon Medical School)
Title: A note on the Dirac operator on symmetric spaces
Abstract: We develop the spectral analysis for the Dirac operator on symmetric spacesof noncompact type based on harmonic analysis on Lie groups. In harmonic analysis onLie groups, the Dirac operator have been deeply studied to characterize discrete series
representations of Lie groups. In this talk, we compute the continuous spectrum of theDirac operator on irreducible symmetric spaces of noncompact type. We show that thecontinuous spectrum has a spectral gap if and only if the symmetric space is isomorphic
to a coset space of the special pseudo-unitary group of odd matrix size. Furthermore, wegive a uniform weighted resolvent estimate for the Dirac operator with respect to spectral
parameter.

ZHOU, Xiaodan (Okinawa Institute of Science and Technology)
Title: Horizontal quasiconvex envelope in the Heisenberg group
Abstract: Various notions of convexity of sets and functions in the Heisenberg group
have been studied in the past two decades. In this talk, we focus on the horizontallyquasiconvex (h-quasiconvex) functions in the Heisenberg group. Inspired by the firstorder characterization and construction of quasiconvex envelope by Barron, Goebel andJensen in the Euclidean space, we obtain a PDE approach to construct the h-quasiconvex
envelope for a given function f in the Heisenberg group. In particular, we show theuniqueness and existence of viscosity solutions to a non-local Hamilton-Jacobi equationand iterate the equation to obtain the h-quasiconvex envelope. Relations between hconvex hull of a set and the h-quasiconvex envelopes are also investigated. This is joint
work with Antoni Kijowski (OIST) and Qing Liu (Fukuoka University/OIST).

FUKUSHIMA, Shota (The University of Tokyo)
Title: Propagation of singularities for Schrödinger equations on manifoldswith ends
Abstract: We employ radially homogeneous wavefront sets, which are introduced by
Ito and Nakamura (Amer. J. Math., 2009), in order to investigate the propagation ofsingularities for Schrödinger equations. The main theorem in my talk is that the wavefront
set propagates to the radially homogeneous wavefront set under the time developmentby the Schrödinger equation. The method of the proof is applicable to both cases ofasymptotically conical and hyperbolic manifolds.

INOUE, Hideki (University of Lyon)
Title: Topological Levinson’s theorem with two Hilbert spaces:from finite graphs to insulators
Abstract: Levinson’s theorem is a fundamental result in quantum mechanics, which relates the scattering matrix to the number of bound states of the underlying quantumsystem. Topological methods for this relation has been considerably developed for thelast about ten years. In this talk, we provide a modification of the method for a scatteringproblem with two Hilbert spaces. The model consists of possibly infinitely many conducting wires attached to an interesting quantum system, which is driven by a self-adjoint
operator satisfying a weak gap type condition on the conducting band of the wires. We
give an explicit formula for a component of the wave operators, which acts on the Hilbert
space of the wires. Topological version of Levinson’s theorem is established in the alreadyexisting C*-algebraic framework. This talk is based on a joint work with Johannes Kellendonk (University Lyon 1) and Hermann Schulz-Baldes (University Erlangen-Nuernberg).

MIRANDA, Pablo (Universidad de Santiago de Chile)
Title: Asymptotic behavior of the Spectral Shift Function for a discrete
Dirac type operator in Z2
Abstract: In this talk, we consider a Dirac type operator in the graph Z2 . This is a matrix
difference operator defined on the vertices and edges of Z2 , together with a perturbation
given by a potential that decays at infinity. We are interested in the spectral propertiesof this operator, which we will study through the analysis of the spectral shift function.
Our main theorem describes the asymptotic behavior of this function near the thresholds
in the spectrum.
The main novelty of this work is related to the nature of the thresholds of our model,
for which the spectral shift function has not been studied before. In particular, we consider
parabolic and hyperbolic thresholds as well as Dirac points.
This talk is based on a joint work with Daniel Parra and Georgi Raikov† .

MIYAZAKI, Hayato (Kagawa University)
Title: Long-range scattering for a homogeneous type nonlinear
Schrödinger equation
Abstract: We consider a homogeneous type nonlinear Schrödinger equation with a timedecaying harmonic oscillator. The nonlinearity of the equation is the homogeneous typewith the long-range critical order in the sense of long-time behavior. As for the standard
NLS with the nonlinearity, namely, without the harmonic oscillator, it is known that theasymptotic behavior of solutions is determined by the shape of the nonlinearity. On theother hand, under the gauge-invariant nonlinearity of the critical order, Kawamoto (2021)
proves that the NLS with the harmonic oscillator admits a solution that behaves like afree solution with a logarithmic phase correction.
In this talk, we give a sufficient condition for the shape of the nonlinearity for thatthe homogeneous type NLS with the harmonic oscillator admits a solution that behaves
like a free solution with or without the phase correction. This talk is based on joint workwith Masaki Kawamoto at Ehime University.

NISHII, Yoshinori (Osaka University)
Title: Energy decay for the semilinear wave equations with weakly
dissipative structure
Abstract: We consider the Cauchy problem for semilinear wave equations in two space dimensions. The null condition is a well-known sufficient condition for the small data global
existence and asymptotic free behavior. Since around 2000, several weaker structural conditions than the null condition were studied. The Agemi condition which introduced byHoshiga (2008), Kubo (2007), Katayama-Matsumura-Sunagawa (2015), etc., is one ofthem. Roughly speaking, this includes not only the classical null condition but also nonlinear dissipative terms. In this talk, we show that the energy of the solution decays withtime when the Agemi condition is satisfied but the null condition is violated. This talkis based on the joint work with Hideaki Sunagawa (Osaka City University) and HirokiTerashita.

TARAMA, Daisuke (Ritsumeikan University)
Title: Sub-Riemannian geodesic flows on the spheres
Abstract: This talk deals with the normal geodesic flows with respect to a certain subRiemannian structure on spheres from the viewpoint of Hamiltonian systems. The caseof trivializable rank five sub-Riemannian structures on the seven-dimensional sphere isparticularly studied and its complete integrability is discussed. The talk is based on ajoint work with Wolfram Bauer (Leibniz Univ. Hannover).

PINCHOVER, Yehuda (Technion-Israel Institute of Technology)
Title: Positive solutions of the A-Laplacian with a potential
Abstract: Criticality theory for the A-Laplacian plus a potential in a Morrey space will
be presented. Prototypes of such operators are the p-Laplacian, the (p, A)-Laplacian,and the pseudo-p-Laplacian plus a potential term. In particular, I will discuss the existence, uniqueness, and simplicity of the principal eigenvalue in a bounded domain, theAgmon-Allegretto-Piepenbrink theorem, as well as certain characterizations of criticality
and subcriticality, the removability of isolated singularity, and the existence of positive
minimal Green functions.
This is a joint work with Yongjun Hou (GTIIT) Antti Rasila (GTIIT).

TADANO, Yukihide (Tokyo University of Science)
Title: Continuum limit problem of discrete Schrödinger operators
Abstract:
We consider asymptotic behavors of discrete Schrödinger operators Hh = −∆h + V |hZd
on the square lattice hZd with mesh size h > 0, where ∆h is the discrete Laplacian andV : Rd → R. It is natural that Hh ”converges” to the Schrödinger operator H = −∆ + Vas h → 0, however there are few mathematical studies of continuum limits of Hh . In thistalk we show that, under relatively weak assumptions of the potential V , Hh converges toH in the generalized norm resolvent sense and that the spectrum σ(H) and eigenfunctionsof H can be approximated by those of Hh . This talk is based on a joint work with Shu
Nakamura (Gakushuin University).

ITAKURA, Kyohei (Ritsumeikan University)
Title: Strong radiation condition and stationary scattering theoryfor Stark operators
Abstract: We present the stationary scattering theory for a perturbed one-body Starkoperator. We deal with several topics on the stationary scattering theory: existence andcompleteness of the stationary wave operators, constructing the associated generalized
Fourier transforms, which diagonalize a perturbed Stark operator and characterizationof asymptotics of the generalized eigenfunctions with minimal growth order in terms of
the stationary scattering matrix. A key ingredient of our procedures is the use of an
improved version of radiation condition bounds obtained previously. Such a stronger,
possibly optimal, radiation condition bounds are proved by a new scheme of proof. Thistalk is based on a joint work with Adachi Tadayoshi (Kyoto Univ.), Ito Kenichi (Univ. ofTokyo) and Skibsted Erik (Aarhus Univ.).

Organized by : FUJIIE, S., HIROTA, K., HOSHIRO, T., ISOZAKI, H., ITAKURA, K.,IWASAKI, C., NAGAYASU, S., NOMURA, Y., RICHARD, S., TAIRA, K.,UMEDA, T., WATANABE, T., and YOSHIDA, N.