日時： 2026年 1月 10日 （土） 14:00 -- 16:30
場所： 学習院大学　南4号館2階 205号室

1. 14:00--15:00
講演者: 村松 亮 氏（東京理科大学）

Title：On wave front sets for the solution of Schr\"odinger equations with time-dependent vector potentials

Abstract: Throughout this talk, we will consider several types of wave front sets for the initial value problem of the Schr\"odinger equation with potentials. The wave front set of a solution to an evolution equation describes how its singularities propagate. For Schr\"odinger equations, one can observe a smoothing effect; that is, singularities of the initial data are instantly transported to spatial infinity along the corresponding classical trajectories. We will focus on the time-dependent vector potentials. In this case, it may become delicate to describe the wave front set due to its time-dependence. We will discuss the characterization of wave front sets for solutions of the Schr\"odinger equation using initial data for $C^{\infty}$, $H^s$ and Gabor wave front sets. In particular, in the framework of the Gabor wave front set, we see that the propagation of singularities of the solution in phase space resembles that of solutions to the wave equation. This talk is based on several joint works with Fumihito Abe (J Institute Co., Ltd.) and with Luigi Rodino (University of Torino) and Elena Cordero ( University of Torino).

2. 15:30--16:30
講演者: 神永 正博 氏（東北学院大学）

Title: Double $\delta$-Shell Models: A Solvable Framework for Core–Shell Quantum Dots

Abstract： A resolvent formula for the double spherical $\delta$-shell interaction is presented, extending the single-shell model of Ikebe and Shimada and enabling a complete description of the system's spectral properties, with particular emphasis on bound states. When both shells are attractive, the model behaves as a spherical double well, allowing explicit estimates of tunneling between the wells. When one shell is attractive and the other repulsive, the same framework provides a simplified representation of quantum confinement in semiconductor nanocrystals. The connection with experimental data for CdSe/ZnS (Type I) and CdTe/CdSe (Type II) quantum dots is also outlined. Modeling the two interfaces by $\delta$-interactions reproduces the observed 0.2–0.3 eV blue shift in Type I and a red shift of comparable magnitude in Type II. Despite its simplicity, the double $\delta$-shell model thus serves as an exactly solvable and physically practical caricature of core-shell quantum confinement.

日時：2025 年 7月 12日 （土） 14:00 -- 16:30
場所： 学習院大学　南4号館2階 205号室

1. 14:00--15:00
講演者: 只野 之英 氏（兵庫県立大学）

Title：On the dispersive estimates for the discrete Schr\"odinger equation on a honeycomb lattice

Abstract: The discrete Schr\"odinger operator on a honeycomb lattice is known as a standard model for graphene. We prove that, when the potential is absent, the free propagator associated to the discrete Schr\"odinger operator on a honeycomb lattice has $\ell^1\to\ell^\infty$ dispersive estimates for wave-packets localized near arbitrary quasi-momentum, and that the slowest time-decay is of order $t^{-2/3}$. The proof is obtained by oscillatory integrals on the momentum space, and one of the difficulties comes from the Dirac points, where the dispersion relation has a conical singularity which disables us to employ the general theories of oscillatory integrals such as Varchenko's theorem. This talk is based on a joint work with Younghun Hong (Chung-Ang University) and Changhun Yang (Chungbuk University).

2. 15:30--16:30
講演者: Serge Richard 氏（名古屋大学）

Title: Wave operators: abstract formulas and application to Dirac operators

Abstract： During this talk, we shall firstly review some stationary expressions for the wave operators of an abstract scattering system. Motivated by the resulting formulas, we shall then apply this approach to Dirac operators in R^3. After various unitary transforms, the corresponding wave operators will be seen as 0-th order pseudodifferential operators on the real line. Some applications will finally be sketched. This presentation is based on an on-going collaboration with A. Alexander, A. Carey, G. Levitina, and A. Rennie.

日時：2025 年 6月 7日 （土） 14:00 -- 16:30
場所： 学習院大学　南4号館2階 205号室

1. 14:00--15:00
講演者: 福嶌 翔太 氏（群馬大学）

Title：Construction and analysis of the solution to Laplace equation for perfect conductors with interface resistance

Abstract: There are numerous studies on Laplace equation with continuity of solution and its flux across interfaces. This transmission condition is called ideal interface condition or perfect bonding, and we can regard it as interfaces without interface resistance. On the other hand, recent studies also examine the transmission　condition in which the solution fails to be continuous across the interface, while the flux remains continuous. This transmission condition is called non-ideal interface condition or imperfect bonding, which appears in various fields of natural science and engineering such as heat conduction, biology and composite materials. The discontinuity of the solution across interfaces is controlled by some function on interfaces called interface resistance. In this talk, we construct the solution to this transmission problem for perfect conductors with constant interface resistance by the layer potentials and the Dirichlet-to-Neumann mapping associated with the exterior problem. As an application, we prove the convergence of the solution as the interface resistance tends to zero. This talk will be based on joint work with Yong-Gwan Ji (KIAS) and Hyeonbae Kang (Inha Univ.).

2. 15:30--16:30
講演者: Fabricio Maciá 氏（マドリード工科大学）

Title: Born approximation and Dirichlet-to-Neumann maps

Abstract： We will focus on the inverse problem of recovering the conductivity in a divergence-form elliptic equation from the knowledge of its Dirichlet-to-Neumann map. This is known as the Calderón problem, or Electric Impedance Tomography, and has been intensively studied in the past forty years. This inverse problem is severely ill-posed, which makes the task of designing efficient algorithms to solve it particularly difficult. We will rigorously prove, in the simplified setting of radial conductivities in the euclidean unit ball, the existence of the Born approximation, a function that encodes the whole DtN map and enjoys several interesting qualitative and quantitative approximation properties. We use this function to factorize the inverse problem into a linear (ill-posed but explicit) and a nonlinear (well-posed, Hölder continuous) part. This factorization gives a (partial) characterization of the set of DtN maps and we will show how this can be used to ultimately design efficient algorithms to solve the inverse problem. Our analysis is based on results on inverse spectral theory for Schrödinger operators on the half-line, in particular on the concept of A-amplitude introduced by Barry Simon in 1999. Recent work on extending this analysis non-radial conductivities in the two dimensional case will also be presented.

日時：2025 年 5月 24 日 （土） 14:00 -- 16:30
場所： 学習院大学　南4号館2階 205号室

1. 14:00--15:00
講演者: 板倉 恭平 氏（学習院大学）

Title：On the absence of complex eigenvalue for distorted semiclassical repulsive Hamiltonians

Abstract: We consider semiclassical Schr\"odinger operators with quadratic or sub-quadratic repulsive potential. Our aim is to characterize resonance free domain for such an operator under a certain non-trapping condition. It is well-known that resonances of a certain Schr\"odinger operator $H$ can be defined as complex eigenvalues of complex dilated $H$. In this talk, we introduce a new distortion based on the classical trajectory of the particle subject to a repulsive electric field, and show that under a virial type condition, a kind of non-trapping condition, distorted repulsive Schr\"odinger operator has no eigenvalues close to the real axis for sufficiently small semiclassical parameter.

2. 15:30--16:30
講演者: Heinz Siedentop 氏（ミュンヘン大学）

Title: How Negative Can Anions Be?

Abstract： The excess charge $Q$ of an atom is the number $N$ of electrons that an atom can maximally bind minus the atomic number $Z$. The excess charge conjecture states that $Q\leq 1$. It is only proven for the most elementary non-relativistic models of atoms. However, it is strongly motivated by the experimental fact that doubly charged stable negative ions are unknown and by numerical evidence. I will review the efforts to prove the conjecture from mathematical models of the atom starting with pioneering work of Ruskai, Sigal, and Lieb up to recent work by Schulz. Moreover, I will present a bound on $Q$ for a relativistic density functional. The talk will be based on joint work with Rafael Benguria, Santiago, Chile.