解析学火曜セミナー

過去の記録 ~10/22次回の予定今後の予定 10/23~

開催情報 火曜日 16:50~18:20 数理科学研究科棟(駒場) 128号室
担当者 中村 周, 石毛 和弘, 伊藤 健一
セミナーURL http://www.ms.u-tokyo.ac.jp/seminar/analysis/

過去の記録

2018年10月16日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
土田哲生 氏 (名城大学)
直積型シュレディンガー方程式の正値解の構造 (日本語)
[ 講演概要 ]
遠方で無限大になるポテンシャル関数をもつ1次元シュレディンガー作用素ふたつからなる2次元の直積型のシュレディンガー方程式を考え、マルチンの理論に基づいてマルチン境界とマルチン核を調べる。ポテンシャルの-1/2乗のべきと-3/2乗のべきが遠方で可積分かどうかに依って、正値解の構造が異なることを示す。(村田實氏(東工大)との共同研究)

2018年07月31日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
蘆田聡平 氏 (京都大学)
長距離型N体問題における散乱行列、一般化フーリエ変換及び伝播評価 (日本語)
[ 講演概要 ]
本講演では長距離型ポテンシャルによるN体問題における散乱行列の一般化固有関数の遠方での漸近挙動に基いた定義を与え、そのようにして得られた散乱行列が波動作用素によって得られる散乱行列と等価であることを示す。また、一般化フーリエ変換を非斉次方程式の外向き進行波解の遠方での漸近挙動よって定義し、その共役作用素がポアソン作用素により与えられることを示す。さらに、クラスターが2つである散乱チャネルに対する新しい改良された伝播評価を散乱チャネルに近い概不変部分空間への正射影作用素を用いて与える。

2018年06月26日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
小川卓克 氏 (東北大学)
移流拡散方程式の初期値問題について (日本語)
[ 講演概要 ]
We consider the Cauchy problem of the drift-diffusion system in the whole space. Introducing the scaling critical case, we consider the solvability of the drift-diffusion system in the whole space and give some large time behavior of solutions. This talk is based on a collaboration with Masaki Kurokiba and Hiroshi Wakui.

2018年06月19日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
Rowan Killip 氏 (UCLA)
KdV is wellposed in $H^{-1}$ (English)

2016年12月13日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
Hans Christianson 氏 (North Carolina State University)
Distribution of eigenfunction mass on some really simple domains (English)
[ 講演概要 ]
Eigenfunctions are fundamental objects of study in spectral geometry and quantum chaos. On a domain or manifold, they determine the behaviour of solutions to many evolution type equations using, for example, separation of variables. Eigenfunctions are very sensitive to background geometry, so it is important to understand what the eigenfunctions look like: where are they large and where are they small? There are many different ways to measure what "large" and "small" mean. One can consider local $L^2$ distribution, local and global $L^p$ distribution, as well as restrictions and boundary values. I will give an overview of what is known, and then discuss some very recent works in progress demonstrating that complicated things can happen even in very simple geometric settings.

2016年12月06日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
Horia Cornean 氏 (オールボー大学、デンマーク)
On the trivialization of Bloch bundles and the construction of localized Wannier functions (English)
[ 講演概要 ]
We shall present an introductory lecture on the trivialization of Bloch bundles and its physical implications. Simply stated, the main question we want to answer is the following: given a rank $N\geq 1$ family of orthogonal projections $P(k)$ where $k\in \mathbb{R}^d$, $P(\cdot)$ is smooth and $\mathbb{Z}^d$-periodic, is it possible to construct an orthonormal basis of its range which consists of vectors which are both smooth and periodic in $k$? We shall explain in detail the connection with solid state physics. This is joint work with I. Herbst and G. Nenciu.

2016年11月29日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
庄司 直高 氏 (筑波大学大学院数理物質科学研究科)
Interior transmission eigenvalue problems on manifolds (Japanese)

2016年10月25日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
山根 英司 氏 (関西学院大学理工学部数理科学科)
可積分離散非線型シュレーディンガー方程式の漸近解析 (JAPANESE)

2016年07月12日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
X. P. Wang 氏 (Université de Nantes, France)
Gevrey estimates of the resolvent and sub-exponential time-decay (English)
[ 講演概要 ]
For a class of non-selfadjoint Schrodinger operators satisfying some weighted coercive condition, we prove that the resolvent satisfies the Gevrey estimates at the threshold. As applications, we show that the heat and Schrodinger semigroups decay sub-exponentially in appropriately weighted spaces. We also study compactly supported perturbations of this class of operators where zero may be an embedded eigenvalue.

2016年06月28日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
Georgi Raikov 氏 (The Pontificia Universidad Católica de Chile)
Discrete spectrum of Schr\"odinger operators with oscillating decaying potentials (English)
[ 講演概要 ]
I will consider the Schr\"odinger operator $H_{\eta W} =-\Delta + \eta W$, self-adjoint in $L^2(\re^d)$, $d \geq 1$. Here $\eta$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. I will discuss the asymptotic behaviour of the discrete spectrum of $H_{\eta W}$ near the origin. Due to the irregular decay of $\eta W$, there exist some non semiclassical phenomena; in particular, $H_{\eta W}$ has less eigenvalues than suggested by the semiclassical intuition.

2016年06月21日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
廣川真男 氏 (広島大学大学院工学研究院)
量子 Rabi 模型に対する Hepp-Lieb-Preparata 量子相転移について (Japanese)
[ 講演概要 ]
本講演では、量子相転移の観点から、量子 Rabi 模型を考察する。Preparata は Hepp-Lieb 量子相転移の数学的構造に基づき、物質と光の相互作用が強くなると、物質・光相互作用系の基底状態が、量子状態の緩和で本来放射すべき光子を纏い始め非摂動論的になることを主張した (Hepp-Lieb-Preparata 量子相転移)。最近、情報通信研究機構の吉原らの回路量子電磁気学の実験で、Hepp-Lieb-Preparata 量子相転移を期待させる結果が得られた。そこで、所謂、A2 項 (光の場の2乗の項) の問題を含め、吉原らが実験で扱った量子 Rabi 模型をHepp-Lieb-Preparata 量子相転移の観点から数理物理学的考察を行う。

2016年06月14日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
新國裕昭 氏 (前橋工科大学)
Schr¥"odinger operators on a periodically broken zigzag carbon nanotube (Japanese)

2016年04月26日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
松原 宰栄 氏 (東大数理)
On microlocal analysis of Gauss-Manin connections for boundary singularities (Japanese)

2016年04月12日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
Jussi Behrndt 氏 (Graz University of Technology)
Scattering matrices and Dirichlet-to-Neumann maps (English)
[ 講演概要 ]
In this talk we discuss a recent result on the representation of the scattering matrix in terms of an abstract Titchmarsh-Weyl m-function. The general result can be applied to scattering problems for Schrödinger operators with $\delta$-type interactions on curves and  hypersurfaces, and scattering problems involving Neumann and Robin realizations of Schrödinger operators on unbounded domains. In both applications we obtain formulas for the corresponding scattering matrices in terms of Dirichlet-to-Neumann maps. This talk is based on joint work with Mark Malamud and Hagen Neidhardt.

2016年01月05日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
Eric Skibsted 氏 (Aarhus University, Denmark)
Stationary scattering theory on manifolds (English)
[ 講演概要 ]
We present a stationary scattering theory for the Schrödinger operator on Riemannian manifolds with the structure of ends each of which is equipped with an escape function (for example a convex distance function). This includes manifolds with ends modeled as cone-like subsets of the Euclidean space and/or the hyperbolic space. Our results include Rellich’s theorem, the limiting absorption principle, radiation condition bounds, the Sommerfeld uniqueness result, and we give complete characterization/asymptotics of the generalized eigenfunctions in a certain Besov space and show asymptotic completeness (with K. Ito).

2015年12月01日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
Stéphane Malek 氏 (Université de Lille, France)
On complex singularity analysis for some linear partial differential equations
[ 講演概要 ]
We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc outside some singular set S. The coefficients are written as linear combinations of powers of a solution X of some first order nonlinear partial differential equation following an idea :we have initiated in a previous joint work with C. Stenger. The solutions Y are shown to develop singularities along the singular set S with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series. (Joint work with A. Lastra and C. Stenger).

2015年11月24日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
許 本源 氏 (東大数理)
A local analysis of the swirling flow to the axi-symmetric Navier-Stokes equations near a saddle point and no-slip flat boundary (English)
[ 講演概要 ]
As one of the violent flow, tornadoes occur in many place of the world. In order to reduce human losses and material damage caused by tornadoes, there are many research methods. One of the effective methods is numerical simulations.  The swirling structure is significant both in mathematical analysis and the numerical simulations of tornado. In this joint work with H. Notsu and T. Yoneda we try to clarify the swirling structure. More precisely, we do numerical computations on axi-symmetric Navier-Stokes flows with no-slip flat boundary. We compare a hyperbolic flow with swirl and one without swirl and observe that the following phenomenons occur only in the swirl case: The distance between the point providing the maximum velocity magnitude $|v|$ and the $z$-axis is drastically changing around some time (which we call it turning point). An ``increasing velocity phenomenon'' occurs near the boundary and the maximum value of $|v|$ is obtained near the axis of symmetry and the boundary when time is close to the turning point.

2015年10月20日(火)

16:50-18:20   数理科学研究科棟(駒場) 128号室
FMSPレクチャーズと共催/部屋が普段と異なりますのでご注意ください
Danielle Hilhorst 氏 (CNRS / University of Paris-Sud)
Existence of an entropy solution in the sense of Young measures for a first order conservation law with a stochastic source term (ENGLISH)
[ 講演概要 ]
We consider a finite volume scheme for a first order conservation law with a monotone flux function and a multiplicative source term involving a Q-Wiener process. We define a stochastic entropy solution in the sense of Young measures. We present some a priori estimates for the discrete solution including a weak BV estimate. After performing a time interpolation, we prove two entropy inequalities and show that the discrete solution converges along a subsequence to an entropy solution in the sense of Young measures.
This is joint work with T. Funaki, Y. Gao and H. Weber.
[ 講演参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Hilhorst151020.pdf

2015年10月13日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
David Sauzin 氏 (CNRS, France)
Nonlinear analysis with endlessly continuable functions (joint work with Shingo Kamimoto) (English)
[ 講演概要 ]
We give estimates for the convolution products of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power series.

2015年09月29日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
Otto Liess 氏 (University of Bologna, Italy)
On the Phragmén-Lindelöf principle for holomorphic functions and factor classes of higher order complex forms in several complex variables
[ 講演概要 ]
In this talk we will discuss maximum principles in unbounded domains in one or several complex variables. We will mainly be interested in these principles for plurisubharmonic (in the one-dimensional case, "subharmonic") or holomorphic functions, when the principles are of
Phragmen-Lindel{\"o}f principle (henceforth called "PL") type. It will turn out that for 2 or more complex variables it will be useful to study our principles together with associated principles for factor classes of complex (0,q) forms with growth type conditions at infinity.
In this abstract we only say something concerning the case of functions. We consider then an open set U in C^n in one or several complex variables. We assume that we are given two real-valued continuous functions f and g on U. We say that PL holds for plurisubharmonic (respectively for holomorphic) functions, if the following implication is true for every plurisubharmonic function $ \rho $ (respectively for every $ \rho $ of form log |h| with h holomorphic) on U: if we know that $ \rho \leq f$ on the boundary of U and if $ (\rho - f)$ is bounded on U, then it must follow that $ \rho \leq g$ on U. ($\rho \leq f$ on the boundary has the following meaning: for ever z in the boundary of U and for every sequence of points y_j in U which tends to z, we have limsup (\rho - f)(y_j) leq 0.) A trivial condition under which PL is true, is when there exists a plurisubharmonic function u on U such that
(*) -g(z) \leq u(z) \leq - f(z) for every z in U.
In fact, if such a function exists, then we can apply the classical maximal principle for unbounded domains to the function $ \rho'= \rho+u$ to obtain at first $ \rho' \leq 0$ and then $ \rho \leq - u \leq g$. It is one of the main goals of the talk to explain how far (*) is from being also a necessary condition for PL. Some examples are intended to justify our approach and applications will be given to problems in convex analysis.

2015年09月08日(火)

16:50-18:20   数理科学研究科棟(駒場) 126号室
水谷治哉 氏 (大阪大学・理学研究科)
長距離型斥力ポテンシャルを持つシュレディンガー方程式の時間大域的ストリッカーツ評価 (Japanese)
[ 講演概要 ]
We will discuss a resent result on global-in-time Strichartz estimates for Schr\"odinger equations with slowly decreasing repulsive potentials. If the potential is of very short-range type, there is a simple method due to Rodnianski-Schlag or Burq et al, which seems to be difficult to apply for the present case. The proof instead follows a similar line as in speaker’s resent joint work with J.-M. Bouclet. In particular, we employ both Morawetz type estimates and the methods of micro local analysis such as the Isozaki-Kitada parametrix, even in the low frequency regime.

2015年07月21日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室
蘆田 聡平 氏 (京都大学理学研究科)
Born-Oppenheimer approximation for an atom in constant magnetic fields (Japanese)
[ 講演概要 ]
We obtain a reduction scheme for the study of the quantum evolution of an atom in constant magnetic fields using the method developed by Martinez, Nenciu and Sordoni based on the construction of almost invariant subspace. Martinez and Sordoni also dealt with such a case but their reduced Hamiltonian includes the vector potential terms. Using the center of mass coordinates and constructing the almost invariant subspace different from theirs, we obtain the reduced Hamiltonian which does not include the vector potential terms. Using the reduced evolution we also obtain the asymptotic expantion of the evolution for a specific localized initial data, which verifies the straight motion of an atom in constatnt magnetic fields.

2015年07月14日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室
Li Yutian 氏 (Department of Mathematics, Hong Kong Baptist University)
Small-time Asymptotics for Subelliptic Heat Kernels (English)
[ 講演概要 ]
Subelliptic operators are the natural generalizations of the Laplace- Beltrami operators, and they play important roles in geometry, several complex variables, probability and physics. As in the classical spectral theory for the elliptic operators, some geometrical properties of the induced subRiemannian geometry can be extracted from the analysis of the heat kernels for subelliptic operators. In this talk we shall review the recent progress in the heat kernel asymptotics for subelliptic operators. We concentrate on the small-time asymptotics of the heat kernel on the diagonal, or equivalently, the asymptotics for the trace. Our interest is to find the exact form of the leading term, and this will lead to a Weyl’s asymptotic formula for the subelliptic operators. This is a joint work with Professor Der-Chen Chang.

2015年05月12日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室
高棹 圭介 氏 (東京大学大学院数理科学研究科)
Brakkeの平均曲率流に対する制約条件付きAllen-Cahn方程式の収束について (Japanese)
[ 講演概要 ]
In this talk we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.

2015年04月21日(火)

16:30-18:00   数理科学研究科棟(駒場) 126号室
今年度から部屋が126に変わりました.
松原 宰栄 氏 (東京大学大学院数理科学研究科)
留数カレントと定数係数線形遅延微分方程式系の一般論について (Japanese)
[ 講演概要 ]
We introduce the ring of differential operators with constant coefficients and commensurate time lags (we use the terminology D$\Delta$ operators from now) initially defined by H. Gl\"using-L\"ur\ss en for ordinary $D\Delta$ operators and observe that various function modules enjoy good cohomological properties over this ring. %After revising the notion of the residue current in the spirit of M. Andersson and E. Wulcan, we introduce the multidimensional version of the ring D$\Delta$ operators.
Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last chapter, we prove the injectivity property of various function modules over this ring as well as spectral synthesis type theorems for $D\Delta$ equations.

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