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解析学火曜セミナー
過去の記録 ～04/26｜次回の予定｜今後の予定 04/27～
| 開催情報 | 火曜日 16:00～17:30 数理科学研究科棟(駒場) 156号室 |
|---|---|
| 担当者 | 石毛 和弘, 坂井 秀隆, 伊藤 健一 |
| セミナーURL | https://www.ms.u-tokyo.ac.jp/seminar/analysis/ |
過去の記録
2013年11月19日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
Alexander Pushnitski 氏 (King's Colledge London)
Inverse spectral problem for positive Hankel operators (ENGLISH)
Alexander Pushnitski 氏 (King's Colledge London)
Inverse spectral problem for positive Hankel operators (ENGLISH)
[ 講演概要 ]
Hankel operators are given by (infinite) matrices with entries
$a_{n+m}$ in $\\ell^2$. We consider inverse spectral problem
for bounded self-adjoint positive Hankel operators.
A famous theorem due to Megretskii, Peller and Treil asserts
that such operators may have any continuous spectrum of
multiplicity one or two and any set of eigenvalues of multiplicity
one. However, more detailed questions of inverse spectral
problem, such as the description of isospectral sets, have never
been addressed. In this talk I will describe in detail the
direct and inverse spectral problem for a particular sub-class
of positive Hankel operators. The talk is based on joint work
with Patrick Gerard (Paris, Orsay).
Hankel operators are given by (infinite) matrices with entries
$a_{n+m}$ in $\\ell^2$. We consider inverse spectral problem
for bounded self-adjoint positive Hankel operators.
A famous theorem due to Megretskii, Peller and Treil asserts
that such operators may have any continuous spectrum of
multiplicity one or two and any set of eigenvalues of multiplicity
one. However, more detailed questions of inverse spectral
problem, such as the description of isospectral sets, have never
been addressed. In this talk I will describe in detail the
direct and inverse spectral problem for a particular sub-class
of positive Hankel operators. The talk is based on joint work
with Patrick Gerard (Paris, Orsay).
2013年07月09日(火)
16:30-18:00 数理科学研究科棟(駒場) 118号室
Tom\'as Lungenstrass 氏 (Pontificia Universidad Catolica de Chile)
A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian
(Joint work with Georgi Raikov) (ENGLISH)
Tom\'as Lungenstrass 氏 (Pontificia Universidad Catolica de Chile)
A Trace Formula for Long-Range Perturbations of the Landau Hamiltonian
(Joint work with Georgi Raikov) (ENGLISH)
[ 講演概要 ]
The Landau Hamiltonian describes the dynamics of a two-dimensional
charged particle subject to a constant magnetic field. Its spectrum
consists in eigenvalues of infinite multiplicity given by $B(2q+1)$, $q\\in Z_+$. We
consider perturbations of this operator by including a continuous
electric potential that decays slowly at infinity (as $|x|^{-\\rho}$, $0<\\rho<1$).
The spectrum of the perturbed operator consists of eigenvalue clusters
which accumulate to the Landau levels. We provide estimates for the
rate at which the clusters shrink as we move up the energy levels.
Further, we obtain an explicit description of the asymptotic density
of eigenvalues for asymptotically homogeneous long-range potentials in
terms of a mean-value transform of the associated homogeneous
function.
The Landau Hamiltonian describes the dynamics of a two-dimensional
charged particle subject to a constant magnetic field. Its spectrum
consists in eigenvalues of infinite multiplicity given by $B(2q+1)$, $q\\in Z_+$. We
consider perturbations of this operator by including a continuous
electric potential that decays slowly at infinity (as $|x|^{-\\rho}$, $0<\\rho<1$).
The spectrum of the perturbed operator consists of eigenvalue clusters
which accumulate to the Landau levels. We provide estimates for the
rate at which the clusters shrink as we move up the energy levels.
Further, we obtain an explicit description of the asymptotic density
of eigenvalues for asymptotically homogeneous long-range potentials in
terms of a mean-value transform of the associated homogeneous
function.
2013年05月21日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
上坂 正晃 氏 (東大数理)
Homogenization in a Thin Layer with an Oscillating Interface and Highly Contrast Coefficients (JAPANESE)
上坂 正晃 氏 (東大数理)
Homogenization in a Thin Layer with an Oscillating Interface and Highly Contrast Coefficients (JAPANESE)
[ 講演概要 ]
We consider the homogenization problem of the elliptic boundary value problem in a thin domain which has a high and low conductivity zones. In our model, two media are separated by a highly oscillating interface. The asymptotic behavior is governed by the order of the thickness of the domain, oscillation period of the interface and contrast between two media. In this talk, we show that the limit problem is changed by these parameters. We also introduce the two-scale convergence result in a thin domain which is the key ingredient of the proof.
We consider the homogenization problem of the elliptic boundary value problem in a thin domain which has a high and low conductivity zones. In our model, two media are separated by a highly oscillating interface. The asymptotic behavior is governed by the order of the thickness of the domain, oscillation period of the interface and contrast between two media. In this talk, we show that the limit problem is changed by these parameters. We also introduce the two-scale convergence result in a thin domain which is the key ingredient of the proof.
2012年12月11日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
講演中止になりました.
Rafe Mazzeo 氏 (Stanford University)
This talk was cancelled! (JAPANESE)
講演中止になりました.
Rafe Mazzeo 氏 (Stanford University)
This talk was cancelled! (JAPANESE)
2012年12月04日(火)
16:30-18:30 数理科学研究科棟(駒場) 128号室
Alexander Vasiliev 氏 (Department of Mathematics, University of Bergen, Norway) 16:30-17:30
Evolution of smooth shapes and the KP hierarchy (ENGLISH)
Group of diffeomorphisms of the unit circle and sub-Riemannian geometry (ENGLISH)
Alexander Vasiliev 氏 (Department of Mathematics, University of Bergen, Norway) 16:30-17:30
Evolution of smooth shapes and the KP hierarchy (ENGLISH)
[ 講演概要 ]
We consider a homotopic evolution in the space of smooth
shapes starting from the unit circle. Based on the Loewner-Kufarev
equation we give a Hamiltonian formulation of this evolution and
provide conservation laws. The symmetries of the evolution are given
by the Virasoro algebra. The 'positive' Virasoro generators span the
holomorphic part of the complexified vector bundle over the space of
conformal embeddings of the unit disk into the complex plane and
smooth on the boundary. In the covariant formulation they are
conserved along the Hamiltonian flow. The 'negative' Virasoro
generators can be recovered by an iterative method making use of the
canonical Poisson structure. We study an embedding of the
Loewner-Kufarev trajectories into the Segal-Wilson Grassmannian,
construct the tau-function, the Baker-Akhiezer function, and finally,
give a class of solutions to the KP hierarchy, which are invariant on
Loewner-Kufarev trajectories.
Irina Markina 氏 (Department of Mathematics, University of Bergen, Norway) 17:30-18:30We consider a homotopic evolution in the space of smooth
shapes starting from the unit circle. Based on the Loewner-Kufarev
equation we give a Hamiltonian formulation of this evolution and
provide conservation laws. The symmetries of the evolution are given
by the Virasoro algebra. The 'positive' Virasoro generators span the
holomorphic part of the complexified vector bundle over the space of
conformal embeddings of the unit disk into the complex plane and
smooth on the boundary. In the covariant formulation they are
conserved along the Hamiltonian flow. The 'negative' Virasoro
generators can be recovered by an iterative method making use of the
canonical Poisson structure. We study an embedding of the
Loewner-Kufarev trajectories into the Segal-Wilson Grassmannian,
construct the tau-function, the Baker-Akhiezer function, and finally,
give a class of solutions to the KP hierarchy, which are invariant on
Loewner-Kufarev trajectories.
Group of diffeomorphisms of the unit circle and sub-Riemannian geometry (ENGLISH)
[ 講演概要 ]
We consider the group of sense-preserving diffeomorphisms of the unit
circle and its central extension - the Virasoro-Bott group as
sub-Riemannian manifolds. Shortly, a sub-Riemannian manifold is a
smooth manifold M with a given sub-bundle D of the tangent bundle, and
with a metric defined on the sub-bundle D. The different sub-bundles
on considered groups are related to some spaces of normalized
univalent functions. We present formulas for geodesics for different
choices of metrics. The geodesic equations are generalizations of
Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDEs. We
show that any two points in these groups can be connected by a curve
tangent to the chosen sub-bundle. We also discuss the similarities and
peculiarities of the structure of sub-Riemannian geodesics on infinite
and finite dimensional manifolds.
We consider the group of sense-preserving diffeomorphisms of the unit
circle and its central extension - the Virasoro-Bott group as
sub-Riemannian manifolds. Shortly, a sub-Riemannian manifold is a
smooth manifold M with a given sub-bundle D of the tangent bundle, and
with a metric defined on the sub-bundle D. The different sub-bundles
on considered groups are related to some spaces of normalized
univalent functions. We present formulas for geodesics for different
choices of metrics. The geodesic equations are generalizations of
Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDEs. We
show that any two points in these groups can be connected by a curve
tangent to the chosen sub-bundle. We also discuss the similarities and
peculiarities of the structure of sub-Riemannian geodesics on infinite
and finite dimensional manifolds.
2012年11月06日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
Thierry Ramond 氏 (Univ. Paris, Orsay)
Resonance free domains for homoclinic orbits (ENGLISH)
Thierry Ramond 氏 (Univ. Paris, Orsay)
Resonance free domains for homoclinic orbits (ENGLISH)
2012年10月30日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
Francis Nier 氏 (Univ. Rennes 1)
About the method of characteristics (ENGLISH)
Francis Nier 氏 (Univ. Rennes 1)
About the method of characteristics (ENGLISH)
[ 講演概要 ]
While studying the mean field dynamics of a systems of bosons, one is led to solve a transport equation for a probability measure in an infinite dimensional phase-space. Since those probability measures are characterized after testing with cylindrical or polynomial observables, which make classes which are not invariant after composing with a nonlinear flow. Thus the standard method of characteristics for transport equations cannot be extended at once to the infinite dimensional case. A solution comes from techniques developed for optimal transport and a probabilistic interpretation of trajectories.
While studying the mean field dynamics of a systems of bosons, one is led to solve a transport equation for a probability measure in an infinite dimensional phase-space. Since those probability measures are characterized after testing with cylindrical or polynomial observables, which make classes which are not invariant after composing with a nonlinear flow. Thus the standard method of characteristics for transport equations cannot be extended at once to the infinite dimensional case. A solution comes from techniques developed for optimal transport and a probabilistic interpretation of trajectories.
2012年10月23日(火)
16:30-18:00 数理科学研究科棟(駒場) 002号室
Elliott Lieb 氏 (Princeton Univ.)
Topics in quantum entropy and entanglement (ENGLISH)
Elliott Lieb 氏 (Princeton Univ.)
Topics in quantum entropy and entanglement (ENGLISH)
[ 講演概要 ]
Several recent results on quantum entropy and the uncertainty
principle will be discussed. This is partly joint work with Eric Carlen
on lower bounds for entanglement, which has no classical analog, in terms
of the negative of the conditional entropy, S1 - S12, whose negativity,
when it occurs, also has no classical analog. (see arXiv:1203.4719)
It is also partly joint work with Rupert Frank on the uncertaintly
principle for quantum entropy which compares the quantum von Neumann
entropy with the classical entropies with respect to two different
bases. We prove an extension to the product of two and three spaces, which
has applications in quantum information theory. (see arxiv:1204.0825)
Several recent results on quantum entropy and the uncertainty
principle will be discussed. This is partly joint work with Eric Carlen
on lower bounds for entanglement, which has no classical analog, in terms
of the negative of the conditional entropy, S1 - S12, whose negativity,
when it occurs, also has no classical analog. (see arXiv:1203.4719)
It is also partly joint work with Rupert Frank on the uncertaintly
principle for quantum entropy which compares the quantum von Neumann
entropy with the classical entropies with respect to two different
bases. We prove an extension to the product of two and three spaces, which
has applications in quantum information theory. (see arxiv:1204.0825)
2012年07月17日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
菅 徹 氏 (東北大学)
2次元円環領域におけるLiouville-Gel\\'fand方程式の非球対称解の構造 (JAPANESE)
菅 徹 氏 (東北大学)
2次元円環領域におけるLiouville-Gel\\'fand方程式の非球対称解の構造 (JAPANESE)
[ 講演概要 ]
指数関数を非線形項に持つ非線形楕円型方程式(Liouville-Gel'fand方程式)について考察する。特に2次元の円環領域では、この方程式の非球対称な解が球対称解から分岐する形で現れる。本講演では、この分岐解の分岐図上での大域的な構造に関して得られた結果を紹介する。
指数関数を非線形項に持つ非線形楕円型方程式(Liouville-Gel'fand方程式)について考察する。特に2次元の円環領域では、この方程式の非球対称な解が球対称解から分岐する形で現れる。本講演では、この分岐解の分岐図上での大域的な構造に関して得られた結果を紹介する。
2012年07月10日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
牛越 惠理佳 氏 (東北大学)
Hadamard variational formula for the Green function
of the Stokes equations with the boundary condition (JAPANESE)
牛越 惠理佳 氏 (東北大学)
Hadamard variational formula for the Green function
of the Stokes equations with the boundary condition (JAPANESE)
[ 講演概要 ]
本講演では、遅い非圧縮粘性流体の運動を記述したStokes方程式におけるHadamard変分公式の導出について考察する。ここで、Hadamard変分公式とは、1908年にHadamardによって提唱された、領域にある摂動をさせた時に、Green函数に代表される領域に依存する函数が、どのような摂動をするのかを表現したものである。
変分公式は、領域摂動に伴う固有値の漸近挙動を明示的に表現したWeylの漸近公式と関連があるなど、領域摂動問題において大変有用なものとして導出されてきた。本講演において、まずは導入として、楕円型方程式で最も基本的なラプラス方程式のDirichlet問題に対する変分公式について言及し、その後に主結果であるStokes方程式におけるHadamard変分公式について述べる。
本講演では、遅い非圧縮粘性流体の運動を記述したStokes方程式におけるHadamard変分公式の導出について考察する。ここで、Hadamard変分公式とは、1908年にHadamardによって提唱された、領域にある摂動をさせた時に、Green函数に代表される領域に依存する函数が、どのような摂動をするのかを表現したものである。
変分公式は、領域摂動に伴う固有値の漸近挙動を明示的に表現したWeylの漸近公式と関連があるなど、領域摂動問題において大変有用なものとして導出されてきた。本講演において、まずは導入として、楕円型方程式で最も基本的なラプラス方程式のDirichlet問題に対する変分公式について言及し、その後に主結果であるStokes方程式におけるHadamard変分公式について述べる。
2012年06月26日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
伊藤 健一 氏 (筑波大学)
Absence of embedded eigenvalues for the Schr\\"odinger operator on manifold with ends (JAPANESE)
伊藤 健一 氏 (筑波大学)
Absence of embedded eigenvalues for the Schr\\"odinger operator on manifold with ends (JAPANESE)
[ 講演概要 ]
We consider a Riemannian manifold with, at least, one expanding end, and prove the absence of $L^2$-eigenvalues for the Schr\\"odinger operator above some critical value. The critical value is computed from the volume growth rate of the end and the potential behavior at infinity. The end structure is formulated abstractly in terms of some convex function, and the examples include asymptotically Euclidean and hyperbolic ends. The proof consists of a priori superexponential decay estimate for eigenfunctions and the absence of superexponentially decaying eigenfunctions, both of which employs the Mourre-type commutator argument. This talk is based on the recent joint work with E.Skibsted (Aarhus University).
We consider a Riemannian manifold with, at least, one expanding end, and prove the absence of $L^2$-eigenvalues for the Schr\\"odinger operator above some critical value. The critical value is computed from the volume growth rate of the end and the potential behavior at infinity. The end structure is formulated abstractly in terms of some convex function, and the examples include asymptotically Euclidean and hyperbolic ends. The proof consists of a priori superexponential decay estimate for eigenfunctions and the absence of superexponentially decaying eigenfunctions, both of which employs the Mourre-type commutator argument. This talk is based on the recent joint work with E.Skibsted (Aarhus University).
2012年05月22日(火)
16:30-18:00 数理科学研究科棟(駒場) 118号室
この日だけ部屋を118に変更します.
Norbert Pozar 氏 (Graduate School of Mathematical Sciences, The University of Tokyo)
Viscosity solutions for nonlinear elliptic-parabolic problems (ENGLISH)
この日だけ部屋を118に変更します.
Norbert Pozar 氏 (Graduate School of Mathematical Sciences, The University of Tokyo)
Viscosity solutions for nonlinear elliptic-parabolic problems (ENGLISH)
[ 講演概要 ]
We introduce a notion of viscosity solutions for a general class of
elliptic-parabolic phase transition problems. These include the
Richards equation, which is a classical model in filtration theory.
Existence and uniqueness results are proved via the comparison
principle. In particular, we show existence and stability properties
of maximal and minimal viscosity solutions for a general class of
initial data. These results are new even in the linear case, where we
also show that viscosity solutions coincide with the regular weak
solutions introduced in [Alt&Luckhaus 1983]. This talk is based on a
recent work with Inwon Kim.
We introduce a notion of viscosity solutions for a general class of
elliptic-parabolic phase transition problems. These include the
Richards equation, which is a classical model in filtration theory.
Existence and uniqueness results are proved via the comparison
principle. In particular, we show existence and stability properties
of maximal and minimal viscosity solutions for a general class of
initial data. These results are new even in the linear case, where we
also show that viscosity solutions coincide with the regular weak
solutions introduced in [Alt&Luckhaus 1983]. This talk is based on a
recent work with Inwon Kim.
2012年05月15日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
水谷 治哉 氏 (京都大学・数理解析研究所)
Strichartz estimates for Schr\\"odinger equations with variable coefficients and unbounded electromagnetic potentials (JAPANESE)
水谷 治哉 氏 (京都大学・数理解析研究所)
Strichartz estimates for Schr\\"odinger equations with variable coefficients and unbounded electromagnetic potentials (JAPANESE)
[ 講演概要 ]
In this talk we consider the Cauchy problem for Schr\\"odinger equations with variable coefficients and unbounded potentials. Under the assumption that the Hamiltonian is a long-range perturbation of the free Schr\\"odinger operator, we construct an outgoing parametrix for the propagator near infinity, and give applications to sharp Strichartz estimates. The basic idea is to combine the standard approximation by using a time dependent modifier, which is not in the semiclassical regime, with the semiclassical approximation of Isozaki-Kitada type. We also show near sharp Strichartz estimates without asymptotic conditions by using local smoothing effects.
In this talk we consider the Cauchy problem for Schr\\"odinger equations with variable coefficients and unbounded potentials. Under the assumption that the Hamiltonian is a long-range perturbation of the free Schr\\"odinger operator, we construct an outgoing parametrix for the propagator near infinity, and give applications to sharp Strichartz estimates. The basic idea is to combine the standard approximation by using a time dependent modifier, which is not in the semiclassical regime, with the semiclassical approximation of Isozaki-Kitada type. We also show near sharp Strichartz estimates without asymptotic conditions by using local smoothing effects.
2012年02月14日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
Michael Loss 氏 (Georgia Institute of Technology)
Symmetry results for Caffarelli-Kohn-Nirenberg inequalities (ENGLISH)
Michael Loss 氏 (Georgia Institute of Technology)
Symmetry results for Caffarelli-Kohn-Nirenberg inequalities (ENGLISH)
2012年01月31日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
Michel Chipot 氏 (University of Zurich)
Obstacle problems in unbounded domains (ENGLISH)
Michel Chipot 氏 (University of Zurich)
Obstacle problems in unbounded domains (ENGLISH)
[ 講演概要 ]
We will present a formulation of obstacle problems in unbounded
domains when the energy method does not work, i.e. whenthe force does not belong to H^{-1}.
We will present a formulation of obstacle problems in unbounded
domains when the energy method does not work, i.e. whenthe force does not belong to H^{-1}.
2011年12月20日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
Gueorgui Raykov 氏 (Catholic University of Chile)
A trace formula for the perturbed Landau Hamiltonian (ENGLISH)
Gueorgui Raykov 氏 (Catholic University of Chile)
A trace formula for the perturbed Landau Hamiltonian (ENGLISH)
[ 講演概要 ]
The talk will be based on a joint work with A. Pushnitski
and C. Villegas-Blas, the preprint is available here:
http://arxiv.org/abs/1110.3098 .
The talk will be based on a joint work with A. Pushnitski
and C. Villegas-Blas, the preprint is available here:
http://arxiv.org/abs/1110.3098 .
2011年12月13日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
Wolfram Bauer 氏 (ゲッチンゲン大学)
Trivializable subriemannian structures and spectral analysis of associated operators (ENGLISH)
Wolfram Bauer 氏 (ゲッチンゲン大学)
Trivializable subriemannian structures and spectral analysis of associated operators (ENGLISH)
2011年11月08日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
寺澤 祐高 氏 (東京大数理（日本学術振興会特別研究員PD）)
確率的摂動項を持つ冪乗法則流体方程式の弱解の存在と 一意性について (JAPANESE)
寺澤 祐高 氏 (東京大数理（日本学術振興会特別研究員PD）)
確率的摂動項を持つ冪乗法則流体方程式の弱解の存在と 一意性について (JAPANESE)
[ 講演概要 ]
本講演では、非圧縮性非ニュートン流体の運動を記述する
偏微分方程式に加法的確率的摂動項を加えた確率偏微分方程式 の弱解
の存在と一意性について考察する。 非ニュートン流体としては、粘性
が変形速度テンソルの大きさの冪乗 の形で依存する冪乗法則流体を考
察し、確率的摂動項としては 有色雑音を考察する。 Necas-Malek-
Ruzicka('93)において、確率的外力項を伴わない、 決定方程式に関し
て示された弱解の存在と一意性の主張を、 確率的摂動項を持つ方程式
に対して示す。 解の存在の証明は、ガレルキン近似によって得られた
解の列に対して、 伊藤の公式、Birkholder-Davis-Gundyの不等式など
により、 解の列のコンパクト性を示すこと及び、解の部分列が収束し
、 その収束先が方程式を弱い意味で満たすことを示すことでなされる。
なお、本講演は、吉田伸生氏(京都大学)との共同研究に基づく。
本講演では、非圧縮性非ニュートン流体の運動を記述する
偏微分方程式に加法的確率的摂動項を加えた確率偏微分方程式 の弱解
の存在と一意性について考察する。 非ニュートン流体としては、粘性
が変形速度テンソルの大きさの冪乗 の形で依存する冪乗法則流体を考
察し、確率的摂動項としては 有色雑音を考察する。 Necas-Malek-
Ruzicka('93)において、確率的外力項を伴わない、 決定方程式に関し
て示された弱解の存在と一意性の主張を、 確率的摂動項を持つ方程式
に対して示す。 解の存在の証明は、ガレルキン近似によって得られた
解の列に対して、 伊藤の公式、Birkholder-Davis-Gundyの不等式など
により、 解の列のコンパクト性を示すこと及び、解の部分列が収束し
、 その収束先が方程式を弱い意味で満たすことを示すことでなされる。
なお、本講演は、吉田伸生氏(京都大学)との共同研究に基づく。
2011年10月11日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
和田出 秀光 氏 (早稲田大学(日本学術振興会特別研究員PD))
重み付きTrudinger-Moser型不等式の最良定数に関して (JAPANESE)
和田出 秀光 氏 (早稲田大学(日本学術振興会特別研究員PD))
重み付きTrudinger-Moser型不等式の最良定数に関して (JAPANESE)
[ 講演概要 ]
同講演では、斉次重み付きTrudinger-Moser型不等式を
最良定数と共に考察する。
重みなしの場合は、 Adachi -Tanaka, Proc. Amer. Math. Soc. (1999),
により全空間上のスケール不変なTrudinger-Moser型不等式が
最良定数と共に導出されており、我々は重み付きTrudinger-Moser型
不等式への拡張を試みる。
更に、重み付きTrudinger-Moser型不等式の偏微分方程式への
応用として、重み付き指数型非線形項を伴うKlein-Gordon方程式を
2次元で考察し、同方程式の局所解及び大域解の存在を証明する。
同講演では、斉次重み付きTrudinger-Moser型不等式を
最良定数と共に考察する。
重みなしの場合は、 Adachi -Tanaka, Proc. Amer. Math. Soc. (1999),
により全空間上のスケール不変なTrudinger-Moser型不等式が
最良定数と共に導出されており、我々は重み付きTrudinger-Moser型
不等式への拡張を試みる。
更に、重み付きTrudinger-Moser型不等式の偏微分方程式への
応用として、重み付き指数型非線形項を伴うKlein-Gordon方程式を
2次元で考察し、同方程式の局所解及び大域解の存在を証明する。
2011年07月12日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
小林 政晴 氏 (東京理科大学)
The inclusion relation between Sobolev and modulation spaces (JAPANESE)
小林 政晴 氏 (東京理科大学)
The inclusion relation between Sobolev and modulation spaces (JAPANESE)
[ 講演概要 ]
In this talk, we consider the inclusion relations between the $L^p$-Sobolev spaces and the modulation spaces. As an application, we give mapping properties of unimodular Fourier multiplier $e^{i|D|^\\alpha}$ between $L^p$-Sobolev spaces and modulation spaces.
Joint work with Mitsuru Sugimoto (Nagoya University).
In this talk, we consider the inclusion relations between the $L^p$-Sobolev spaces and the modulation spaces. As an application, we give mapping properties of unimodular Fourier multiplier $e^{i|D|^\\alpha}$ between $L^p$-Sobolev spaces and modulation spaces.
Joint work with Mitsuru Sugimoto (Nagoya University).
2011年04月26日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
片岡 清臣 氏 (東京大学大学院数理科学研究科)
On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)
片岡 清臣 氏 (東京大学大学院数理科学研究科)
On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)
2010年11月16日(火)
16:00-18:30 数理科学研究科棟(駒場) 123号室
GCOE miniworkshopと合同
打越敬祐 氏 (防衛大学) 16:00-16:45
Hyperfunctions and vortex sheets (ENGLISH)
L. Boutet de Monvel 氏 (University of Paris 6) 17:00-18:30
Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)
GCOE miniworkshopと合同
打越敬祐 氏 (防衛大学) 16:00-16:45
Hyperfunctions and vortex sheets (ENGLISH)
L. Boutet de Monvel 氏 (University of Paris 6) 17:00-18:30
Residual trace and equivariant asymptotic trace of Toeplitz operators (ENGLISH)
2010年09月28日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
Pavel Exner 氏 (Czech Academy of Sciences)
Some spectral and resonance properties of quantum graphs (ENGLISH)
Pavel Exner 氏 (Czech Academy of Sciences)
Some spectral and resonance properties of quantum graphs (ENGLISH)
[ 講演概要 ]
In this talk I will discuss three new results about Schr¨odinger operators
on metric graphs obtained in collaboration with Jiri Lipovskyand Brian Davies.
The first one is related to invalidity of the uniform continuation principle for such
operators. One manifestation of this fact are embedded eigenvalues due to
rational relations of graph edge lengths. This effect is non-generic and we show
how geometric perturbations turn these embedded eigenvalues into resonances.
Then second problem is related to high-energy behavior of resonances: we extend
a recent result of Davies and Pushnitski to graphs with general vertex couplings
and find conditions under which the asymptotics does not have Weyl character.
Finally, the last question addressed here concerns the absolutely continuous spectrum
of radial-tree graphs. In a similar vein, we generalize a recent result by Breuer and
Frank that in case of the free (or Kirhhoff) coupling the ac spectrum is absent
provided the edge length are increasing without a bound along the tree.
We show that the result remains valid for a large class of vertex couplings,
but on the other hand, there are nontrivial couplings leading to an ac spectrum.
In this talk I will discuss three new results about Schr¨odinger operators
on metric graphs obtained in collaboration with Jiri Lipovskyand Brian Davies.
The first one is related to invalidity of the uniform continuation principle for such
operators. One manifestation of this fact are embedded eigenvalues due to
rational relations of graph edge lengths. This effect is non-generic and we show
how geometric perturbations turn these embedded eigenvalues into resonances.
Then second problem is related to high-energy behavior of resonances: we extend
a recent result of Davies and Pushnitski to graphs with general vertex couplings
and find conditions under which the asymptotics does not have Weyl character.
Finally, the last question addressed here concerns the absolutely continuous spectrum
of radial-tree graphs. In a similar vein, we generalize a recent result by Breuer and
Frank that in case of the free (or Kirhhoff) coupling the ac spectrum is absent
provided the edge length are increasing without a bound along the tree.
We show that the result remains valid for a large class of vertex couplings,
but on the other hand, there are nontrivial couplings leading to an ac spectrum.
2010年07月13日(火)
17:00-18:00 数理科学研究科棟(駒場) 128号室
Carlos Villegas Blas 氏 (メキシコ国立自治大学)
On a limiting eigenvalue distribution theorem for perturbations of the hydrogen atom (JAPANESE)
Carlos Villegas Blas 氏 (メキシコ国立自治大学)
On a limiting eigenvalue distribution theorem for perturbations of the hydrogen atom (JAPANESE)
[ 講演概要 ]
Let H be the hydrogen atom Hamiltonian. We will show that
the operator H+P can have well defined clusters of eigenvalues
for a suitable perturbation P=f(h)Q where Q is a pseudo-differential
operator of order zero and f(h) is a small quantity depending of
the Planck's parameter h. We will show that the distribution of
eigenvalues in those clusters has a semi-classical limit involving
the averages of the principal symbol of Q along the classical orbits
of the Kepler problem.
Let H be the hydrogen atom Hamiltonian. We will show that
the operator H+P can have well defined clusters of eigenvalues
for a suitable perturbation P=f(h)Q where Q is a pseudo-differential
operator of order zero and f(h) is a small quantity depending of
the Planck's parameter h. We will show that the distribution of
eigenvalues in those clusters has a semi-classical limit involving
the averages of the principal symbol of Q along the classical orbits
of the Kepler problem.
2010年06月22日(火)
16:30-18:00 数理科学研究科棟(駒場) 128号室
Ivana Alexandrova 氏 (East Carolina University)
Resonances for Magnetic Scattering by Two Solenoidal Fields at Large Separation (ENGLISH)
Ivana Alexandrova 氏 (East Carolina University)
Resonances for Magnetic Scattering by Two Solenoidal Fields at Large Separation (ENGLISH)
[ 講演概要 ]
We consider the problem of quantum resonances in magnetic scattering by two
solenoidal fields at large separation in two dimensions, and we study how a trajectory
oscillating between the two fields gives rise to resonances near the real axis when
the distance between two centers of fields goes to infinity. We give a sharp lower
bound on resonance widths in terms of backward amplitudes calculated explicitly for
scattering by each solenoidal field. The study is based on a new type of complex
scaling method. As an application, we also discuss the relation to semiclassical
resonances in scattering by two solenoidal fields. This is joint work with Hideo Tamura.
We consider the problem of quantum resonances in magnetic scattering by two
solenoidal fields at large separation in two dimensions, and we study how a trajectory
oscillating between the two fields gives rise to resonances near the real axis when
the distance between two centers of fields goes to infinity. We give a sharp lower
bound on resonance widths in terms of backward amplitudes calculated explicitly for
scattering by each solenoidal field. The study is based on a new type of complex
scaling method. As an application, we also discuss the relation to semiclassical
resonances in scattering by two solenoidal fields. This is joint work with Hideo Tamura.
