seminar0401

平成27年度(2015 Apr. --2016 Mar.)

第150回学習院大学スペクトル理論セミナー

日時：　2016 年 2月 27 日（土） 14:30 -- 17:30
場所：学習院大学　南4号館2階 205セミナー室

1. 14:30--15:30
講師: 檀　裕也氏(松山大学)
題目: TBA

tea time 15:30--16:00

2. 16:00--17:30
講師: Heinz Siedentop氏 (University of Munich)
題目: Ground State Energy of Heavy Atoms: The Leading Correction

追加情報

学習院早稲田幾何学セミナーでは以下のような講演があります.
解析学的な話題です.

2015年12月7日(月）16:00--17:30
Speaker： Richard Schoen (UC Irvine, Stanford University)
Title: Extremal eigenvalue problems and minimal surfaces

http://www.math.gakushuin.ac.jp/~hosono/GWGS/GWGS.html

第149回学習院大学スペクトル理論セミナー

日時：　2015 年 12月 5 日（土） 14:00 -- 18:00
場所：学習院大学　南4号館2階 205セミナー室

1. 14:00--15:00
講師: Dhriti Ranjan Dolai 氏 (Chennai Mathematical Institute)
題目: Towards integrated density of state for decaying randomness

概略: abstract

---15:00-15:30 tea time ---

2. 15:30--16:30
講師: Vit Jakubsky氏 (Department of Theoretical Physics Institute of Nuclear Physics, Czeck Rep.)
題目: Sufficient conditions for confinement of Dirac fermions in graphene

概略: We study the confinement of Dirac fermions in graphene and in carbon
nanotubes by external magnetic fields, mechanical deformations or
inhomogeneities in the substrate. By applying variational principles to the
square of the Dirac operator, we obtain sufficient and necessary conditions
for existence of discrete energy levels, i.e. confinement of the
quasi-particles. The rigorous theoretical results are illustrated on the
realistic examples of the three classes of traps.

---16:30-17:00 tea time ---

3. 17:00--18:00
講師: Christian Sadel氏 ( IST Austria )
題目: Complex analytic one-frequency cocycles
(joint with A. Avila and S. Jitomirskaya)

概略:
Given an analytic function A(x) from the one-dimensional Torus into the set of
d by d complex matrices and given a frequency a one can define a cocycle map
f(x,v)=(x+a, A(x)v) where x is an element of the Torus and v a vector.
Iterating the map leads to a one-frequency cocycle, the dynamical behavior is
characterized by Lyapunov exponents. We show joint continuity of Lyapunov
exponents at irrational frequencies, give a characterization for the existence
of a dominated splitting and prove that for a fixed irrational frequency and
a dense subset of analytic maps $A(x)$ the cocycle is dominated.

第148回学習院大学スペクトル理論セミナー

日時：　2015 年 11月 28 日（土） 15:00 -- 16:00
場所：学習院大学　南4号館2階 205セミナー室

講師: Frederic Klopp 氏(Université Pierre et Marie Curie)
題目: Stark-Wannier ladders and cubic exponential sums

概略: The talk is devoted to Stark-Wannier ladders
i.e. the resonances of a one dimensional periodic operator in
a constant electric field. These periodic sequences of points
in the lower half of the complex plane have been conjecture to
be very sensitive to the number theoretical properties of the
electric field. Computing the asymptotics of the reflection
coefficients in the case of a simple 1-periodic potential, we
related the resonances to cubic exponential sums in which the
frequency is computed from the electric field. In the case of
rational frequency, we derive "large imarginary part"
asymptotics for the resonances. The talk is based on joint
work with A. Fedotov (St Petersburg).

集中講義のお知らせ(12月1日--12月3日, 12月4日--12月5日)

集中講義の部屋は全て、南 4-205　です.

・12月1日--12月3日

　講師：Frederic Klopp 氏(Université Pierre et Marie Curie)
　題目：Interacting one-dimensional quantum particles in a random field

　概略: Consider $N$ one-dimensional quantum particles in
　　　　　the interval $[0,L]$ submitted to a random field. The
　　　　　interaction is assumed to be repulsive. In the thermodynamic
　　　　　limit ($L,N\to+\infty$, $N/L\to\rho>0$), the aim of the
　　　　　lectures is to describe the ground state and the ground state
　　　　　energy per particle when $\rho$ is small. We will discuss both
　　　　　the bosonic and the fermionic cases.

　12月1日（火）15:00--17:00
　Lecture 1: we will give an introduction to the problem and
　discuss the main results wit hout proofs.

　12月2日（水）17:00--19:00
　Lecture 2: we will present and at least partially prove novel
　results on the one particle hamiltonian that are preparatory
　to the study of the interacting particles. These results will
　describe both the low lying spectrum and the associated
　eigenfunctions. We will also explain how these result lead to
　a reduced model for interacting particles.

　12月3日（木）15:00 --17:00
　Lecture 3: we will present our analysis of the reduced model
　and explain how it leads to the main results presented in lecture 1.

・12月4日--12月5日

　講師：Christopher Shirley (Universite Libre de Bruxelles)
　題目：　Decorrelation estimates for random operators

　概略: abstract

　１２月４日（金）15:00--17:00
　１２月５日（土）10:00--12:00

小谷眞一先生（大阪大学名誉教授）の、
１０月２３日（金）～２８日（水）の集中講義と
スペクトル理論セミナーのご案内.

・集中講義
１０月２３日（金）１回目　午後３時～５時
１０月２６日（月）２回目　午後３時～５時
１０月２７日（火）３回目　午後５時～７時
１０月２８日（水）４回目　午前１０時～１２時

タイトル：1Dシュレーディンガー作用素とKdV方程式
－無反射性をkey wordにしてー

アブストラクト：1Dシュレーディンガー作用素とKdV方程式などの可積分系との関係に
ついては1960年代以来よく知られている。この講義では無反射性を持つ初期関数（振動
する関数）の空間に対して佐藤-Segal-Wilson理論を適用することによりKdV方程式の解
を力学系として構成する。また解の極限的な性質についても考察する。

第147回学習院大学スペクトル理論セミナー
日時：　2015 年 10月 24 日（土） 14:30 -- 17:00
場所：学習院大学　南4号館2階 205セミナー室

1. 14:30--15:30
講師: 佐々木格氏(信州大学)
題目: 相対論的シュレディンガー作用素の埋蔵固有値について

概略: 遠方で減衰するポテンシャルV(x)をもつ相対論的シュレディンガー作用素で正の
固有値を持つものが存在することを示す。ポテンシャルは滑らかかつ遠方で
1/|x|のオーダーで減衰する。

15:30-- 16:00 tea time

2. 16:00--17:00
講師: 小谷眞一氏（大阪大学名誉教授）
題目: エルゴード的1Dシュレーディンガー作用素のスペクトルについて

概略: エルゴード的1Dシュレーディンガー作用素は、周期的ポテンシャルか
ら概周期的あるいはランダムなポテンシャルを含む広いクラスの振動するポテンシャル
をもつシュレーディンガー作用素である。セミナーではその基礎理論と最近の話題に
ついて概Mathieu作用素、Fibonacciポテンシャルを例にしながら解説する。

第146回学習院大学スペクトル理論セミナー

日時：　2015 年 9月 26 日（土） 14:30 -- 17:00
場所：学習院大学　南4号館2階 205セミナー室

1. 14:30--15:30
講師: 貝塚公一氏 (学習院大学)

題目: A characterization of the $L^{2}$-range of the
Poisson transform on symmetric spaces of noncompact type

概略: We develop the stationary scattering theory for a system of invariant
differential operators on symmetric spaces of noncompact type.
When spectral parameter is real and singular, a certain degeneracy arises
in asymptotic behavior of joint eigenfunctions at infinity.
In order to characterize the $L^{2}$-range of the Poisson
transform, we introduce an Agmon-H\"{o}rmander type norm
in accordance with degeneracy of the spectral parameter.
We also prove a scattering formula for joint eigenfunctions which
involves degenerate components.

Tea time

2. 16:00--17:00

講師: Fulton Gonzalez 氏 (Tufts University)

題目: Mean Value Operators, Integral Geometry, and Group Theory

概略: An old classical result says that a continuous function $f$ on
$\mathbb R^n$ is harmonic if and only if its average value over any sphere
equals its value at the center. In this talk, we'll explore various extensions
of this principle, focusing primarily on symmetric spaces.
We will also explore various integral transforms associated with mean
value operators, and consider some applications to wave equations
in one and many time variables.

第145回学習院大学スペクトル理論セミナー

日時：　2015 年 7月 18 日（土） 14:00 -- 17:00
場所：学習院大学　南4号館2階 205セミナー室

1. 14:00--15:00
講師: 清水翔之氏 (東工大理工)

題目: 平均場近似の枠組みにおける量子多体系の散乱理論について

概略:
量子多体系を非線形シュレーディンガー方程式を古典場とする系の量子化と捉え、
その古典極限を考察する.
古典極限のレベルにおいて系の古典軌道からの揺らぎを記述する
生成・消滅に関して二次の（時間依存する）量子場のハミルトニアン
のスペクトル解析において得られた結果を報告する.
本研究は佐々木浩宣氏（千葉大学）、鈴木章斗氏（信州大学）との共同研究に基づく.

tea time 15:00--15:30

2. 15:30--17:00
講師: 川本昌紀氏(神戸大)

題目: パルス磁場上での散乱

概略:
本講演では時間に周期的に on, off を繰り返すパルス磁場上での荷電粒子の漸近挙
動について考察する。
特に、磁場の周期、大きさ、粒子の質量、電荷によって、荷電粒子が束縛状態。等速
直線運動のように散乱する。指数的な速さで散乱する。の３つの状態になりうる事を
見る。さらに、荷電粒子が指数的に散乱する場合では、非常に弱い減衰を持つポテン
シャルにおいても波動作用素が存在し、さらに完全である事を見る。
なお本講演は足立匡義氏(神戸大学)との共同研究に基づくものである。

第144回学習院大学スペクトル理論セミナー

日時：　2015 年 6月 27 日（土） 14:30 -- 17:00
場所：学習院大学　南4号館2階 205セミナー室

1. 14:30--15:30

講師:　Gianfausto Dell’Antonio 氏(SISSA, Trieste)

題目:　Fat graph shrinking to a metric graph: the problem of boundary
conditions at the vertices

Short summary:
A fat graph is a tubular neighborhood of a metric graph with Dirichlet b.c.
at the surface. For a metric graph the dynamics is
determined by the boundary conditions at the vertices. We show, in the case
of a star graph, that, when the radius of the neighborhood tends to zero,
the limit Schroedinger dynamics is controlled by the zero-energy resonances
of the Schroedinger operator on the fat graph (and therefore by the shape of
the fat graph). If there are no resonances, if the limit exists in strong
resolvent convergence it has Dirichlet b.c. by a detailed analysis of the
case of the half line.

tea time

2. 16:00--17:00

講師: Trinh Khanh Duy氏（九州大学IMI）

題目: Spectral measures of random Jacobi matrices and associated
Hermite polynomials.

Abstract: Consider a random (symmetric) Jacobi matrix consisting of two
i.i.d. sequences of random variables, one for the diagonal and the other
for the off diagonal. The common distributions of the diagonal and the
off diagonal are standard Gaussian distribution and the square root of
the Gamma distribution with parameters $(1, \alpha), \alpha > 0$,
respectively. The matrix is regarded as the limit of Gaussian beta
ensembles (G$\beta$E for short) as the matrix sizes $N$ tend to infinity
with the constraint that $N\beta=2\alpha$. We investigate spectral
measures of the random Jacobi matrix and obtain the explicit formula for
the mean spectral measure. It turns out that the mean spectral measure
coincides with the probability measure of the associated Hermite
polynomials.

第143回学習院大学スペクトル理論セミナー

日時：　2015 年 6月 20 日（土） 14:30 -- 17:00
場所：学習院大学　南4号館2階 205セミナー室

1. 14:30--15:30

講師：　蘆田聡平氏（京都大学理学研究科D1）

題目：　Born-Oppenheimer approximation for an atom in constant magnetic
fields

概略：
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.
In Martinez-Sordoni such a case is also studied 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.

15:30--16:00 tea time

2. 16:00--17:00

講師：　Gianfausto Dell’Antonio 氏(SISSA, Trieste)

題目： Quadratic form for the contact interaction of three fermions

概略:
We give the quadratic form for the contact interaction of two identical
fermions of unit mass and a third particle of mass m.
We give the ranges of m for which the corresponding symmetric operator has
only one self-adjoint extension, the range for which there is a one-parameter
family of extensions, the one for which all extensions are unbounded below.
We also show that when there is only one extension it can be seen as limit,
in strong resolvent convergence, of a sequence of Hamiltonians with a regular
two-body potential which have a zero energy resonance

第142回学習院大学スペクトル理論セミナー

日時：　2015 年 5月 16 日（土） 14:30 -- 17:00
場所：学習院大学　南4号館2階 205セミナー室

1. 14:30—15:30
講師: 高津飛鳥氏(首都大学東京　理工学研究科)

題目: Riemannian Wasserstein geometry on the space of Gaussian measures
over the Wiener space

概略:
An abstract Wiener space is a triple $(H,E, \mu)$,
where the Cameron--Martin space $H$ is a separable Hilbert space,
the Wiener measure $\mu$ plays as a standard Gaussian measure on $H$,
and $E$ is a separable Banach space on which $\mu$ is supported.
It is known that the space of Gaussian measures on $E$ being
equivalent to $\mu$ becomes a Hilbert manifold,
and the manifold admits a non-positive Riemannian metric, so-called
the Fisher metric due to the information geometry.
In this talk, we construct a non-negative Riemannian metric,
whose Riemannian distance function coincides with the Wasserstein
distance function.
The Wasserstein distance function is a (pseudo) distance function on
the space of probability measures, whose induced topology is weaker
than the weak topology.
We explain the difference/relation between two geometries
and we give some applications using both geometries.

This is based on a joint work with Hiroshi KAWABI(Okayama University)

tea time

2. 16:00—17:00
講師: Andr\’{e} Martinez氏(University of Bologna)

題目: Width of resonances for Helmholtz resonators with straight neck in
any dimension.

概略: This is a joint work with Alain Grigis and Thomas Duykaerts
where we obtain an optimal bound on the width of the lowest resonance for
a general Helmholtz resonators with straight neck. Such a resonator
consists of a bounded cavity, connected with the exterior by a thin tube.
The frequency of the sounds it produces are determined by the shape of
the cavity, and one expects that their duration is related to the length
of the tube and to the diameter of its section. From a mathematical point
of view, this phenomena is described by the resonances of the Dirichlet
Laplacian on the domain consisting of the union of the cavity, the tube,
and the exterior. Here, we obtain an asymptotic estimate on the duration
of the sounds when the width of the tube tends to zero. This estimate
shows that this duration is exponentially large, with a rate that is
proportional to the length of the tube, and inversely proportional to its
width. The proof is based on previous results obtained by Hislop-Martinez
and Martinez-N\’{e}d\’{e}lec, and on a recent version of Carleman
estimates due to Kenig-Sj\”{o}strand-Uhlman.