Himeji Conference on Partial Differential Equations
Titles and abstracts 2024

MINE, Takuya (Kyoto Institute of Technology)
Title: Asymptotics of IDS for the random point interactions
Abstract: In this talk, we give a survey about the asymptotics of the integrated densityof states (IDS) at the bottom of the spectrum for Schrödinger operators with random
point interactions. Especially, we prove that IDS for 2 or 3-dimensional Poisson pointinteractions decays polynomially at −∞, and verify it by a numerical method.

TAGAWA, Tomoya (The University of Tokyo)
Title: Low energy LAP for slowly decaying attractive potentials
Abstract: In this talk, based on a joint work with Kenichi Ito, we discuss the uniform
resolvent estimates near zero energy for the Schrödinger operator
H = − 12 ∆ + V + q on Rd
with d ∈ N = {1, 2, . . .}. The potential V is slowly decaying and attractive, and q
is a short-range perturbation relative to V . More precisely, we assume the following
conditions. We use notation ⟨x⟩ = (1 + x2 )1/2 .
Condition. Let V ∈ C 2 (Rd ) be spherically symmetric, and there exist ν, ϵ ∈ (0, 2) andc, C > 0 such that for any |α| ≤ 2 and x ∈ Rd|∂ α V (x)| ≤ C⟨x⟩−ν−|α| ,V (x) ≤ −c⟨x⟩−ν ,x · (∇V (x)) ≤ −(2 − ϵ)V (x).

In addition, let q ∈ L∞ (Rd ), and there exist ν ′ ∈ (ν, 2] and C ′ > 0 such that for anyx ∈ Rd′|q(x)| ≤ C ′ ⟨x⟩−1−ν /2 .
Remark: Nakamura (1994), Fournais-Skibsted (2004), Richard (2006) and Skibsted (2013)
discussed the LAP for C ∞ slowly decaying attractive potentials. We extend these previous
results to C 2 potentials in the framework of an appropriate Agmon-Hörmander space
without using microlocal analysis.
To begin with, we introduce Rellich’s theorem which asserts the absence of generalized
eigenfunctions with nonnegative eigenvalues in a certain weighted space called the AgmonHörmander spaces. Rellich’s theorem is an important basis for our theory. Next, we will
introduce the limiting absorption principle bounds and the radiation condition bounds.
From these two estimates, we obtain the LAP. Finally, the Sommerfeld type uniquenesstheorem, which characterizes the limiting resolvents, is introduced. In the proofs of ourtheorems, we mainly use a commutator argument recently invented by Ito-Skibsted.

BOURNE, Chris (Nagoya University)
Title: Spectral and topological properties of quantum walks
Abstract: Quantum walks, quantum analogues of random walks, bring together aspects
of probability theory, quantum information theory and mathematical physics. In thistalk, I will introduce an operator algebraic method to compute the essential spectrumof (discrete-time) quantum walk unitaries. For quantum walks with additional symmetries, we can also apply techniques from index theory to characterise ‘topologically stable’
bound states. This is partly based on arXiv:2211.10601 as well as work-in-progress withS. Richard and Y. Tanaka.

NATSUME, Toshikazu
Title: A secondary index for certain non-Fredholm operators
Abstract: We study a family of non-Fredholm operators. These operators are associatedwith the shift operator on the binary tree. We show that we can still extract numerical
data so that operators are distinguished from each other. This is joint work with R. Nest.

BANERJEE, Rudrajit (Okinawa Institute of Science and Technology)
Title: From Riemannian to Lorentzian manifolds: complex metrics
and the Wick rotated heat semigroup
Abstract: We introduce a ‘Wick rotation’, parameterized by θ ∈ [0, π/2], that interpolates
between Riemannian and Lorentzian metrics on a smooth real manifold admitting codimension one foliations. Defined using the foliation structure, this interpolation maps
between a Riemannian metric at θ = π/2 and a Lorentzian metric at θ = 0, passing
through complex metrics at intermediate values of θ. We prove that for each θ ∈ (0, π/2],
the associated Laplace-Beltrami operator generates an analytic semigroup, the ‘Wickrotated heat semigroup’, extending the usual heat semigroup arising from the Dirichlet
Laplacian on a Riemannian manifold. In particular, for all θ ∈ (0, π/2] : (i) the Wickrotated heat semigroup is smoothing on L2 , (ii) it has a smooth integral kernel, and (iii)the kernel’s diagonal admits a small semigroup time asymptotic expansion. Finally, weshow that (with an additional hypothesis) the Wick rotated heat semigroup converges toa unitary Schrödinger evolution group as θ → 0, i.e. in the strict Lorentzian limit.
This talk is based on joint work with Max Niedermaier at the University of Pittsburgh.

MIYAO, Tadahiro (Hokkaido University)
Title: Stability of charge density waves in electron-phonon systems
Abstract: Theoretical physicists have suggested that various types of order, such as
superconducting and ferromagnetic states, emerge in many-electron systems on lattices.
However, the mathematical justification for these phenomena is still not fully understood.
In this presentation, I will outline mathematical results concerning one of the typicalorders in many-electron systems, namely, charge density waves. Firstly, I will briefly review the rigorous results on charge density waves in many-electron systems in two or moredimensions by C. Borgs and R. Kotecky. Subsequently, I will explain the stabilization ofcharge density waves in systems where electrons interact with lattice vibrations. This result aligns with predictions made by theoretical physicists using numerical computations.
The key points of the proof are as follows: 1. Extension of the Pirogov-Sinai theory
by Borgs et al. for fermionic systems to systems where fermions interact with bosons.
2. Application of explicit formulas for correlation functions revealed in the rigorous studyof quantum field theory to control contributions from bosons to the partition function.

NAKAHASHI, Wataru (Tokyo University of Science)
Title: Non-Smoothness of the fundamental solutions for Schrödinger equations
with super-quadratic and radially symmetric potentials
Abstract: We consider the non-smoothness of the fundamental solution for the Schrödingerequation with a radially symmetric and super-quadratic potential. Yajima (1996) hasshown that if the potential is super-quadratic and the spatial dimension is one, then thefundamental solution E(t, x, y) is not smooth anywhere with respect to (t, x, y). Yajima
has conjectured that the same result is true even for higher dimensional cases. In this talk,we show that the fundamental solution E(t, x, y) does not belong to C 1 as a function of(t, x, y) if the potential is radially symmetric and super-quadratic, the spatial dimension
is larger than or equals to 3. This talk is based on a joint work with Keiichi Kato (TokyoUniversity of Science) and Yukihide Tadano (Tokyo University of Science).

ISHIDA, Akari (Nagoya University)
Title: A depth-dependent stability estimate in an iterative method
for solving a Cauchy problem for the Laplace equation
Abstract: In this talk, we consider the Cauchy problem for the Laplace operator. We
construct approximate solutions by using the iterative method proposed by Bastay, Kozlov
and Turesson. In this method, we solve the corresponding boundary value problems
repeatedly. Then, we show that we construct them more stably when we choose thesmaller domain where we consider the boundary value problems. Furthermore, since thismethod works with inexact data, we also deal with this case.

YOSHIDA, Naoya (Ritsumeikan University)
Title: Trace asymptotics for the Iwatsuka Hamiltonian with strong electric
field
Abstract: In this talk, we consider the 2D magnetic Schrödinger operator
P = Dx2 + (Dy − b(x))2 + µV (x, y),where V is a non-negative scalar potential decaying at infinity. The magnetic field isassumed to be positive, bounded and monotonically increasing on R.
We obtain a complete asymptotic expansion of trf(P) as the coupling constant µ tends
to +∞, where f ∈ C0∞ (R). We also give a Weyl type asymptotics formula with optimal
reminder estimate of the counting function of eigenvalues of P .
This is based on the joint work with Profs. Shin-ichi Shirai (Osaka Institute of Technology).

PARK, Kanam (National Institute of Technology, Toba College)
Title: Birational representations of affine Weyl groups arising from a Lax form
(1)of the q-Painlevé equation of type E6(1)Abstract: The q-Painlevé equation of type E6 is given as a q-difference system with an(1)affine Weyl group symmetry of type E6 . We obtained a 3 × 3 Lax formalism for theequation as a new one. Our goal is to understand a birational representation of an affine(1)Weyl group symmetry of type E6 as actions on the 3Ö3 Lax formalism. In this talk, wetalk about a relation between it and known birational actions of 3 kinds of affine Weyl(1)group of type A2 as a partial result.

MASAKI, Satoshi (Hokkaido University)
Title: Global existence and asymptotic behavior in nonlinear Schrödinger
systems without null and dissipative structure
Abstract: We consider a class of systems of cubic nonlinear Schrödinger equations withtwo components in one dimension. It is known that cubic nonlinearity is critical in one
dimension in view of the large-time behavior. (cf. modified scattering). Compared tothe single-equation case, systems exhibit a wide range of asymptotic behaviors for theirsolutions. Our motivation is to comprehend the effect of general nonlinearities that are
gauge-invariant and represented by cubic polynomials with real coefficients.
For systems within this class, the global existence of small solutions is not always
guaranteed. One sufficient condition for global existence is the presence of a conserved
quantity equivalent to the (L2 )2 -norm of the solution. In previous studies of large-time
behavior, the assumption of the existence of such a conserved quantity is often made,characterized as a so-called weak null condition. Another sufficient condition for global
existence is the validity of a dissipative condition, indicating the non-increasing propertyof a quantity equivalent to the (L2 )2 -norm of the solution.
In this talk, we introduce a specific class of systems that do not satisfy the weak nullcondition or the dissipative condition, and that, however, we can establish the global
existence of solutions for small data. We also provide an asymptotic profile of such
solutions in terms of a solution to the corresponding ODE system.

Organized by : HIGUCHI, K., HIROTA, K., ITAKURA, K., KAMEOKA, K., NAGAYASU, S.,
RICHARD, S., TAIRA, K., WATANABE, T., YOSHIDA, N.