## Tuesday Seminar of Analysis

Seminar information archive ～09/18｜Next seminar｜Future seminars 09/19～

Date, time & place | Tuesday 16:00 - 17:30 156Room #156 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | ISHIGE Kazuhiro, SAKAI Hidetaka, ITO Kenichi |

**Seminar information archive**

### 2010/09/28

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Some spectral and resonance properties of quantum graphs (ENGLISH)

**Pavel Exner**(Czech Academy of Sciences)Some spectral and resonance properties of quantum graphs (ENGLISH)

[ Abstract ]

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.

### 2010/07/13

17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On a limiting eigenvalue distribution theorem for perturbations of the hydrogen atom (JAPANESE)

**Carlos Villegas Blas**(Universidad Nacional Autonoma de Mexico)On a limiting eigenvalue distribution theorem for perturbations of the hydrogen atom (JAPANESE)

[ Abstract ]

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.

### 2010/06/22

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Resonances for Magnetic Scattering by Two Solenoidal Fields at Large Separation (ENGLISH)

**Ivana Alexandrova**(East Carolina University)Resonances for Magnetic Scattering by Two Solenoidal Fields at Large Separation (ENGLISH)

[ Abstract ]

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.

### 2010/06/15

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Sato's counterexample and the structure of generalized functions (JAPANESE)

**Takashi Takiguchi**(Department of Mathematics, National Defense Academy)Sato's counterexample and the structure of generalized functions (JAPANESE)

[ Abstract ]

In this talk, we discuss the relation between the structure and the microlocal unique continuation property of generalized functions. We also mention some applications of the microlocal unique continuation property.

In this talk, we discuss the relation between the structure and the microlocal unique continuation property of generalized functions. We also mention some applications of the microlocal unique continuation property.

### 2010/04/13

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Strichartz estimates and the Isozaki-Kitada parametrix

on asymptotically hyperbolic manifolds (ENGLISH)

**Jean-Marc Bouclet**(Toulouse University, France)Strichartz estimates and the Isozaki-Kitada parametrix

on asymptotically hyperbolic manifolds (ENGLISH)

### 2010/01/26

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Finite gap Jacobi matrices (joint work with Barry Simon and Maxim Zinchenko)

**Jacob S. Christiansen**(コペンハーゲン大学)Finite gap Jacobi matrices (joint work with Barry Simon and Maxim Zinchenko)

### 2010/01/19

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

超函数の有界性と Massera 型定理について

**岡田 靖則**(千葉大・理)超函数の有界性と Massera 型定理について

### 2009/11/24

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Analytic Properties of Eigen Values of Daubechies Localization Operator

**吉野 邦生**(東京都市大学)Analytic Properties of Eigen Values of Daubechies Localization Operator

[ Abstract ]

1)ドーベシー局在化作用素の固有値の解析的性質、積分表示式、

(2)ドーベシー局在化作用素のシンボルの再現公式、

(3)ドーベシー局在化作用素のバーグマンーフォック空間での表示

等について述べる。

1)ドーベシー局在化作用素の固有値の解析的性質、積分表示式、

(2)ドーベシー局在化作用素のシンボルの再現公式、

(3)ドーベシー局在化作用素のバーグマンーフォック空間での表示

等について述べる。

### 2009/09/15

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

渦層の超局所解析

**打越 敬祐**(防衛大学校数学教育室)渦層の超局所解析

[ Abstract ]

渦層とは,2次元流体が界面を境に2層に分かれて流れる状態で,今井功氏は「佐藤超関数は渦層である」と言っている.この考え方を用いると,

界面の時間変化を記述するBirkoff-Rott方程式を,擬微分方程式に書き直して解けることを説明する.

渦層とは,2次元流体が界面を境に2層に分かれて流れる状態で,今井功氏は「佐藤超関数は渦層である」と言っている.この考え方を用いると,

界面の時間変化を記述するBirkoff-Rott方程式を,擬微分方程式に書き直して解けることを説明する.

### 2009/07/21

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Low Energy Asymptotics of the SSF for Pauli Operators with Non-Constant Magnetic Fields

**Georgi Raikov**(PUC, Chile)Low Energy Asymptotics of the SSF for Pauli Operators with Non-Constant Magnetic Fields

[ Abstract ]

In my talk, I will consider the 3D Pauli operator with non-constant magnetic field of constant direction,

perturbed by a matrix-valued electric potential which decays fast enough at infinity. I will discuss

the low-energy asymptotics of the associated spectral shift function which is proportional to the eigenvalue

counting function at negative energies, and to the scattering phase at positive energies.

In my talk, I will consider the 3D Pauli operator with non-constant magnetic field of constant direction,

perturbed by a matrix-valued electric potential which decays fast enough at infinity. I will discuss

the low-energy asymptotics of the associated spectral shift function which is proportional to the eigenvalue

counting function at negative energies, and to the scattering phase at positive energies.

### 2009/06/30

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

The Structure of the Scattering Amplitude for Schrodinger Operators with a Strong Magnetic Field

**Ivana Alexandrova**(東京大数理)The Structure of the Scattering Amplitude for Schrodinger Operators with a Strong Magnetic Field

[ Abstract ]

We study the microlocal structure of the semi-classical scattering amplitude for Schrodinger operators with a strong magnetic field at non-trapping energies. We prove that, up to any order, the scattering amplitude can be approximated by a semi-classical pseudodifferential-operator-valued Fourier integral operator.

We study the microlocal structure of the semi-classical scattering amplitude for Schrodinger operators with a strong magnetic field at non-trapping energies. We prove that, up to any order, the scattering amplitude can be approximated by a semi-classical pseudodifferential-operator-valued Fourier integral operator.

### 2009/06/02

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

無限階擬微分作用素の形式核関数について

**神本 晋吾**(東京大数理)無限階擬微分作用素の形式核関数について

### 2009/05/26

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Intrepolation problems in H￥"ormander algebras

**Myriam Ounaies**(Strasbourg大学数学科)Intrepolation problems in H￥"ormander algebras

[ Abstract ]

We call Hörmander algebras the spaces $A_p(\\mathbb C)$ of entire functions $f$ such that, for all $z$ in $\\mathbb C$, \\[|f(z)|\\le Ae^{Bp(z)},\\] where $A$ and $B$ are some positive constants (depending on $f$) and $p$ is a subharmonic weight. We consider the following interpolation problem : Given a discrete sequence $\\{a_j\\}$ of complex numbers and a sequence of complex values $\\{b_j\\}$, under what conditions does there exist a function $f\\in A_p(\\mathbb C)$ such that $f(a_j)=b_j$ for all $j$ ? In other words, what is the trace of $A_p(\\mathbb C)$ on $\\{a_j\\}$ ?

We say that $\\{a_j\\}$ is an interpolating sequence if the trace is defined by the space of all $\\{b_j\\}$ satisfying $|b_j|\\le A'e^{B'p(a_j)}$, for some constants $A',B'>0$.

We use Hörmander's $L^2$-estimates for the $\\bar\\partial$-equation to describe the trace when the weight $p$ is radial and doubling and to characterize the interpolating sequences for more general weights.

We call Hörmander algebras the spaces $A_p(\\mathbb C)$ of entire functions $f$ such that, for all $z$ in $\\mathbb C$, \\[|f(z)|\\le Ae^{Bp(z)},\\] where $A$ and $B$ are some positive constants (depending on $f$) and $p$ is a subharmonic weight. We consider the following interpolation problem : Given a discrete sequence $\\{a_j\\}$ of complex numbers and a sequence of complex values $\\{b_j\\}$, under what conditions does there exist a function $f\\in A_p(\\mathbb C)$ such that $f(a_j)=b_j$ for all $j$ ? In other words, what is the trace of $A_p(\\mathbb C)$ on $\\{a_j\\}$ ?

We say that $\\{a_j\\}$ is an interpolating sequence if the trace is defined by the space of all $\\{b_j\\}$ satisfying $|b_j|\\le A'e^{B'p(a_j)}$, for some constants $A',B'>0$.

We use Hörmander's $L^2$-estimates for the $\\bar\\partial$-equation to describe the trace when the weight $p$ is radial and doubling and to characterize the interpolating sequences for more general weights.

### 2009/04/28

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

非線型消散項を伴うシュレディンガー方程式の任意の大きさの初期データに対する解の漸近挙動(北直泰氏との共同研究)

**下村 明洋**(首都大学東京)非線型消散項を伴うシュレディンガー方程式の任意の大きさの初期データに対する解の漸近挙動(北直泰氏との共同研究)

### 2009/01/20

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Generating function of eigenvalues of Daubechies Localization Operator

**吉野 邦生**(武蔵工業大学)Generating function of eigenvalues of Daubechies Localization Operator

[ Abstract ]

Daubechies Localization Operator の 固有値の母関数から symbol 関数を再現する公式について

Daubechies Localization Operator の 固有値の母関数から symbol 関数を再現する公式について

### 2009/01/12

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Finite gap Jacobi matrices (joint work with Barry Simon and Maxim Zinchenko)

**Jacob S. Christiansen**

(コペンハーゲン大学)Finite gap Jacobi matrices (joint work with Barry Simon and Maxim Zinchenko)

### 2008/11/25

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Heat kernels for subelliptic operators

**Ovidiu Calin**(Eastern Michigan University)Heat kernels for subelliptic operators

[ Abstract ]

Subelliptic operators are differential operators with missing

directions. Their behavior is very different than the behavior or

elliptic operators. Among the most well known subelliptic operators

are the Grusin operator, the Heisenberg operator, and the Kolmogorov

operator. There are several methods of finding the heat kernels of

subelliptic operators. The heat kernels of subelliptic operators are

usually represented in integral form, but in the case of the

Kolmogorov operator we shall show that the heat kernel is of function

type. We shall spend some time on other subelliptic operators too.

Subelliptic operators are differential operators with missing

directions. Their behavior is very different than the behavior or

elliptic operators. Among the most well known subelliptic operators

are the Grusin operator, the Heisenberg operator, and the Kolmogorov

operator. There are several methods of finding the heat kernels of

subelliptic operators. The heat kernels of subelliptic operators are

usually represented in integral form, but in the case of the

Kolmogorov operator we shall show that the heat kernel is of function

type. We shall spend some time on other subelliptic operators too.

### 2008/11/11

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Rotation number approach to spectral analysis of the generalized Kronig-Penney Hamiltonians

**新國 裕昭**(首都大学東京)Rotation number approach to spectral analysis of the generalized Kronig-Penney Hamiltonians

### 2008/10/28

17:00-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Introduction to geometric analysis of hyperbolic equations

**Serge Alinhac**(パリ大学オルセイ校)Introduction to geometric analysis of hyperbolic equations

### 2008/10/14

16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Thin 3D Navier-Stokes equations: Ultimate boundedness of solutions with large data and global attractors

**George Sell**(ミネソタ大学)Thin 3D Navier-Stokes equations: Ultimate boundedness of solutions with large data and global attractors

[ Abstract ]

グローバルCOE連続講演会と共催です.詳細はそちらをご覧ください.

グローバルCOE連続講演会と共催です.詳細はそちらをご覧ください.

### 2008/05/20

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Wave operators for diatomic molecules

**Vania Sordoni**(ボローニャ大学)Wave operators for diatomic molecules

### 2008/05/13

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Resonances for non-analytic potentials (joint work with T. Ramond and J. Sj\\"ostrand)

**Andr\'e Martinez**(ボローニャ大学)Resonances for non-analytic potentials (joint work with T. Ramond and J. Sj\\"ostrand)

### 2008/03/13

15:00-17:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Schr/"odinger equations on scattering manifolds and microlocal singularities

Local smoothing in the presence of lots of trapping

[ Reference URL ]

http://agusta.ms.u-tokyo.ac.jp/seminerphotos2/Zworski-abstract.pdf

**伊藤健一**(東京大学大学院数理科学研究科) 15:00-16:00Schr/"odinger equations on scattering manifolds and microlocal singularities

**Maciej ZWORSKI**(カリフォルニア大学バークレイ校) 16:30-17:30Local smoothing in the presence of lots of trapping

[ Reference URL ]

http://agusta.ms.u-tokyo.ac.jp/seminerphotos2/Zworski-abstract.pdf

### 2008/01/22

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Magnetic Schroedinger operators and twisted crossed product

**Serge Richard**(Univ. Lyon 1)Magnetic Schroedinger operators and twisted crossed product

[ Abstract ]

During this seminar, we shall study spectral properties of generalized

magnetic Schroedinger operators H(B,V) with anisotropic magnetic field B

and scalar potential V. The essential spectrum of such operators is

expressed as a union of spectra of some asymptotic operators supported by

the quasi-orbits of a suitable dynamical system. A localization property

of the functional calculus of H(B,V) will also be presented. It directly

implies a non-propagation result for the unitary group generated by this

operator. The proofs rely on the use of twisted crossed product

C*-algebras. Twisted dynamical systems and their corresponding algebras

will be introduced and the natural link with magnetic Schroedinger

operators will be clearly established.

During this seminar, we shall study spectral properties of generalized

magnetic Schroedinger operators H(B,V) with anisotropic magnetic field B

and scalar potential V. The essential spectrum of such operators is

expressed as a union of spectra of some asymptotic operators supported by

the quasi-orbits of a suitable dynamical system. A localization property

of the functional calculus of H(B,V) will also be presented. It directly

implies a non-propagation result for the unitary group generated by this

operator. The proofs rely on the use of twisted crossed product

C*-algebras. Twisted dynamical systems and their corresponding algebras

will be introduced and the natural link with magnetic Schroedinger

operators will be clearly established.

### 2008/01/08

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On the restrictions of Laplace-Beltrami eigenfunctions to curves

**Nikolay Tzvetkov**(Lille大学)On the restrictions of Laplace-Beltrami eigenfunctions to curves