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（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（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.