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Tuesday Seminar of Analysis
Seminar information archive ～04/24｜Next seminar｜Future seminars 04/25～
| 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
2019/11/26
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
ASHIDA Sohei (Gakushuin University)
Accurate lower bounds for eigenvalues of electronic Hamiltonians (Japanese)
ASHIDA Sohei (Gakushuin University)
Accurate lower bounds for eigenvalues of electronic Hamiltonians (Japanese)
[ Abstract ]
Electronic Hamiltonians are differential operators depending on relative positions of nuclei as parameters. When we regard an eigenvalues of an electronic Hamiltonian as a function of relative positions of nuclei, minimum points correspond to shapes of molecules. Upper bounds for eigenvalues are obtained by variational methods. However, since the physical information as minimum points does not change when a reference point of energy changes, physical information can not be obtained by variational methods only. Combining lower and upper bounds physical information is obtained.
In this talk we discuss the Weinstein-Arnszajn intermediate problem methods for lower bounds of eigenvalues based on comparison of operators. A method for lower bounds of one-electronic Hamiltonians is also introduced. Some computations for two kinds of hydrogen molecule-ion are shown.
Electronic Hamiltonians are differential operators depending on relative positions of nuclei as parameters. When we regard an eigenvalues of an electronic Hamiltonian as a function of relative positions of nuclei, minimum points correspond to shapes of molecules. Upper bounds for eigenvalues are obtained by variational methods. However, since the physical information as minimum points does not change when a reference point of energy changes, physical information can not be obtained by variational methods only. Combining lower and upper bounds physical information is obtained.
In this talk we discuss the Weinstein-Arnszajn intermediate problem methods for lower bounds of eigenvalues based on comparison of operators. A method for lower bounds of one-electronic Hamiltonians is also introduced. Some computations for two kinds of hydrogen molecule-ion are shown.
2019/11/19
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Wenjia Jing (Tsinghua University)
Quantitative homogenization for the Dirichlet problem of Stokes system in periodic perforated domain - a unified approach (English)
Wenjia Jing (Tsinghua University)
Quantitative homogenization for the Dirichlet problem of Stokes system in periodic perforated domain - a unified approach (English)
[ Abstract ]
We present a new unified approach for the quantitative homogenization of the Stokes system in periodically perforated domains, that is domains outside a periodic array of holes, with Dirichlet data at the boundary of the holes. The method is based on the (rescaled) cell-problem and is adaptive to the ratio between the typical distance and the typical side length of the holes; in particular, for the critical ratio identified by Cioranescu-Murat, we recover the “strange term from nowhere”termed by them, which, in the context of Stokes system, corresponds to the Brinkman’s law. An advantage of the method is that it can be systematically quantified using the periodic layer potential technique. We will also report some new correctors to the homogenization problem using this approach. The talk is based on joint work with Yong Lu and Christophe Prange.
We present a new unified approach for the quantitative homogenization of the Stokes system in periodically perforated domains, that is domains outside a periodic array of holes, with Dirichlet data at the boundary of the holes. The method is based on the (rescaled) cell-problem and is adaptive to the ratio between the typical distance and the typical side length of the holes; in particular, for the critical ratio identified by Cioranescu-Murat, we recover the “strange term from nowhere”termed by them, which, in the context of Stokes system, corresponds to the Brinkman’s law. An advantage of the method is that it can be systematically quantified using the periodic layer potential technique. We will also report some new correctors to the homogenization problem using this approach. The talk is based on joint work with Yong Lu and Christophe Prange.
2019/11/12
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
KAMIMOTO Shingo (Hiroshima University)
Mould expansion and resurgent structure (Japanese)
KAMIMOTO Shingo (Hiroshima University)
Mould expansion and resurgent structure (Japanese)
2019/11/05
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Ngô Quốc Anh (Vietnam National University, Hanoi / the University of Tokyo)
Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space (English)
Ngô Quốc Anh (Vietnam National University, Hanoi / the University of Tokyo)
Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space (English)
[ Abstract ]
This talk concerns entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[\Delta^m u = \pm u^\alpha\] in $\mathbb R^n$ with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. For small $m$, the above equations arise in many physical phenomena and applied mathematics. They also arise from several prescribing geometric curvture problems in conformal geometry such as the Yamabe problem, the scalar curvature problem, and the Q-curvature problem for the Paneitz operator. Higher-order cases also arise from the Q-curvature problem for the GJMS operator. In this talk, I will present a complete picture of the existence and non-existence of solutions to the above equations in the full rage of the parameters $n$, $m$, and $\alpha$. This is joint work with V.H. Nguyen, Q.H. Phan, and D. Ye.
This talk concerns entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[\Delta^m u = \pm u^\alpha\] in $\mathbb R^n$ with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. For small $m$, the above equations arise in many physical phenomena and applied mathematics. They also arise from several prescribing geometric curvture problems in conformal geometry such as the Yamabe problem, the scalar curvature problem, and the Q-curvature problem for the Paneitz operator. Higher-order cases also arise from the Q-curvature problem for the GJMS operator. In this talk, I will present a complete picture of the existence and non-existence of solutions to the above equations in the full rage of the parameters $n$, $m$, and $\alpha$. This is joint work with V.H. Nguyen, Q.H. Phan, and D. Ye.
2019/06/11
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Antonio De Rosa (Courant Institute of Mathematical Sciences)
Solutions to two conjectures in branched transport: stability and regularity of optimal paths (English)
Antonio De Rosa (Courant Institute of Mathematical Sciences)
Solutions to two conjectures in branched transport: stability and regularity of optimal paths (English)
[ Abstract ]
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $\alpha$ is bigger than a critical threshold, but we prove it for every exponent $\alpha \in (0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely solve a conjecture of the book Optimal transportation networks by Bernot, Caselles and Morel. Moreover the robustness of our proof allows us to get the stability for more general lower semicontinuous functional. Furthermore, we prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the aforementioned book. We use the latter result to show the regularity of the optimal networks. (Joint works with Maria Colombo and Andrea Marchese)
Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. The transportation cost in these models is proportional to a concave power $\alpha \in (0,1)$ of the intensity of the flow. We focus on the stability of the optimal transports, with respect to variations of the source and target measures. The stability was known when $\alpha$ is bigger than a critical threshold, but we prove it for every exponent $\alpha \in (0,1)$ and we provide a counterexample for $\alpha=0$. Thus we completely solve a conjecture of the book Optimal transportation networks by Bernot, Caselles and Morel. Moreover the robustness of our proof allows us to get the stability for more general lower semicontinuous functional. Furthermore, we prove the stability for the mailing problem, which was completely open in the literature, solving another conjecture of the aforementioned book. We use the latter result to show the regularity of the optimal networks. (Joint works with Maria Colombo and Andrea Marchese)
2019/04/09
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Fabio Punzo (Politecnico di Milano)
The Poisson equation on Riemannian manifolds (English)
Fabio Punzo (Politecnico di Milano)
The Poisson equation on Riemannian manifolds (English)
[ Abstract ]
The talk is concerned with the existence of solutions to the Poisson equation on complete non-compact Riemannian manifolds. In particular, the interplay between the Ricci curvature and the behaviour at infinity of the source function will be discussed. This is a joint work with G. Catino and D.D. Monticelli.
The talk is concerned with the existence of solutions to the Poisson equation on complete non-compact Riemannian manifolds. In particular, the interplay between the Ricci curvature and the behaviour at infinity of the source function will be discussed. This is a joint work with G. Catino and D.D. Monticelli.
2019/03/05
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Nicholas Edelen (Massachusetts Institute of Technology)
The structure of minimal surfaces near polyhedral cones (English)
Nicholas Edelen (Massachusetts Institute of Technology)
The structure of minimal surfaces near polyhedral cones (English)
[ Abstract ]
We prove a regularity theorem for minimal varifolds which resemble a cone $C_0$ over an equiangular geodesic net. For varifold classes admitting a ``no-hole'' condition on the singular set, we additionally establish regularity near the cone $C_0 \times R^m$. Our result implies the following generalization of Taylor's structure theorem for soap bubbles: given an $n$-dimensional soap bubble $M$ in $R^{n+1}$, then away from an $(n-3)$-dimensional set, $M$ is locally $C^{1,\alpha}$ equivalent to $R^n$, a union of three half-$n$-planes meeting at $120$ degrees, or an $(n-2)$-line of tetrahedral junctions. This is joint work with Maria Colombo and Luca Spolaor.
We prove a regularity theorem for minimal varifolds which resemble a cone $C_0$ over an equiangular geodesic net. For varifold classes admitting a ``no-hole'' condition on the singular set, we additionally establish regularity near the cone $C_0 \times R^m$. Our result implies the following generalization of Taylor's structure theorem for soap bubbles: given an $n$-dimensional soap bubble $M$ in $R^{n+1}$, then away from an $(n-3)$-dimensional set, $M$ is locally $C^{1,\alpha}$ equivalent to $R^n$, a union of three half-$n$-planes meeting at $120$ degrees, or an $(n-2)$-line of tetrahedral junctions. This is joint work with Maria Colombo and Luca Spolaor.
2019/01/22
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
KATO Keiichi (Tokyo University of Science)
Construction of solutions to Schrodinger equations with sub-quadratic potential via wave packet transform (Japanese)
KATO Keiichi (Tokyo University of Science)
Construction of solutions to Schrodinger equations with sub-quadratic potential via wave packet transform (Japanese)
[ Abstract ]
In this talk, we consider linear Schrodinger equations with sub-quadratic potentials, which can be transformed by the wave packet transform with time dependent wave packet to a PDE of first order with inhomogeneous terms including unknown function and second derivatives of the potential. If the second derivatives of the potentials are bounded, the homogenous term of the first oder equation gives a construction of solutions to Schrodinger equations with sub-quadratic potentials by the similar way as in D. Fujiwara's work for Feynman path integral. We will show numerical computations by using our construction, if we have enough time.
In this talk, we consider linear Schrodinger equations with sub-quadratic potentials, which can be transformed by the wave packet transform with time dependent wave packet to a PDE of first order with inhomogeneous terms including unknown function and second derivatives of the potential. If the second derivatives of the potentials are bounded, the homogenous term of the first oder equation gives a construction of solutions to Schrodinger equations with sub-quadratic potentials by the similar way as in D. Fujiwara's work for Feynman path integral. We will show numerical computations by using our construction, if we have enough time.
2018/12/25
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
MASAKI Satoshi (Osaka University)
Modified scattering for nonlinear dispersive equations with critical non-polynomial nonlinearities (Japanese)
MASAKI Satoshi (Osaka University)
Modified scattering for nonlinear dispersive equations with critical non-polynomial nonlinearities (Japanese)
[ Abstract ]
In this talk, I will introduce resent progress on modified scattering for Schrodinger equation and Klein-Gordon equation with a non-polynomial nonlinearity. We use Fourier series expansion technique to find the resonant part of the nonlinearity which produces phase correction factor.
In this talk, I will introduce resent progress on modified scattering for Schrodinger equation and Klein-Gordon equation with a non-polynomial nonlinearity. We use Fourier series expansion technique to find the resonant part of the nonlinearity which produces phase correction factor.
2018/11/06
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
SHIBATA Tetsutaro (Hiroshima University)
Global behavior of bifurcation curves and related topics (日本語)
SHIBATA Tetsutaro (Hiroshima University)
Global behavior of bifurcation curves and related topics (日本語)
[ Abstract ]
In this talk, we consider the asymptotic behavior of bifurcation curves for ODE with oscillatory nonlinear term. First, we study the global and local behavior of oscillatory bifurcation curves. We also consider the bifurcation problems with nonlinear diffusion.
In this talk, we consider the asymptotic behavior of bifurcation curves for ODE with oscillatory nonlinear term. First, we study the global and local behavior of oscillatory bifurcation curves. We also consider the bifurcation problems with nonlinear diffusion.
2018/10/30
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
MIYANISHI Yoshihisa (Osaka University)
Spectral structure of the Neumann-Poincaré operator in three dimensions: Willmore energy and surface geometry (日本語)
MIYANISHI Yoshihisa (Osaka University)
Spectral structure of the Neumann-Poincaré operator in three dimensions: Willmore energy and surface geometry (日本語)
[ Abstract ]
The Neumann-Poincaré operator (abbreviated by NP) is a boundary integral operator naturally arising when solving classical boundary value problems using layer potentials. If the boundary of the domain, on which the NP operator is defined, is $C^{1, \alpha}$ smooth, then the NP operator is compact. Thus, the Fredholm integral equation, which appears when solving Dirichlet or Neumann problems, can be solved using the Fredholm index theory.
Regarding spectral properties of the NP operator, the spectrum consists of eigenvalues converging to $0$ for $C^{1, \alpha}$ smooth boundaries. Our main purpose here is to deduce eigenvalue asymptotics of the NP operators in three dimensions. This formula is the so-called Weyl's law for eigenvalue problems of NP operators. Then we discuss relationships among the Weyl's law, the Euler characteristic and the Willmore energy on the boundary surface.
The Neumann-Poincaré operator (abbreviated by NP) is a boundary integral operator naturally arising when solving classical boundary value problems using layer potentials. If the boundary of the domain, on which the NP operator is defined, is $C^{1, \alpha}$ smooth, then the NP operator is compact. Thus, the Fredholm integral equation, which appears when solving Dirichlet or Neumann problems, can be solved using the Fredholm index theory.
Regarding spectral properties of the NP operator, the spectrum consists of eigenvalues converging to $0$ for $C^{1, \alpha}$ smooth boundaries. Our main purpose here is to deduce eigenvalue asymptotics of the NP operators in three dimensions. This formula is the so-called Weyl's law for eigenvalue problems of NP operators. Then we discuss relationships among the Weyl's law, the Euler characteristic and the Willmore energy on the boundary surface.
2018/10/16
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
TSUCHIDA Tetsuo (Meijo University)
Positive solutions of Schr\"odinger equations in product form (日本語)
TSUCHIDA Tetsuo (Meijo University)
Positive solutions of Schr\"odinger equations in product form (日本語)
2018/07/31
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
ASHIDA Sohei (Kyoto University)
Scattering matrices, generalized Fourier transforms and propagation estimates in long-range N-body problems (日本語)
ASHIDA Sohei (Kyoto University)
Scattering matrices, generalized Fourier transforms and propagation estimates in long-range N-body problems (日本語)
[ Abstract ]
We give a definition of scattering matrices in long-range N-body problems based on the asymptotic behaviors of generalized eigenfunctions and show that these scattering matrices are equivalent to the ones defined by wave-operator approach. We also define generalized Fourier transforms by the asymptotic behaviors of outgoing solutions to nonhomogeneous equations and show that the adjoint operators of them are given by Poisson operators. We also consider new improved propagation estimates for two-cluster scattering channels using projections onto almost invariant subspaces close to two-cluster scattering channels.
We give a definition of scattering matrices in long-range N-body problems based on the asymptotic behaviors of generalized eigenfunctions and show that these scattering matrices are equivalent to the ones defined by wave-operator approach. We also define generalized Fourier transforms by the asymptotic behaviors of outgoing solutions to nonhomogeneous equations and show that the adjoint operators of them are given by Poisson operators. We also consider new improved propagation estimates for two-cluster scattering channels using projections onto almost invariant subspaces close to two-cluster scattering channels.
2018/06/26
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
OGAWA Takayoshi (Tohoku University)
The Cauchy problem of the drift-diffusion system in R^n (日本語)
OGAWA Takayoshi (Tohoku University)
The Cauchy problem of the drift-diffusion system in R^n (日本語)
[ Abstract ]
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.
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.
2018/06/19
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Rowan Killip (UCLA)
KdV is wellposed in $H^{-1}$ (English)
Rowan Killip (UCLA)
KdV is wellposed in $H^{-1}$ (English)
2016/12/13
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
Hans Christianson (North Carolina State University)
Distribution of eigenfunction mass on some really simple domains (English)
Hans Christianson (North Carolina State University)
Distribution of eigenfunction mass on some really simple domains (English)
[ Abstract ]
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.
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.
2016/12/06
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
Horia Cornean (Aalborg University, Denmark)
On the trivialization of Bloch bundles and the construction of localized Wannier functions (English)
Horia Cornean (Aalborg University, Denmark)
On the trivialization of Bloch bundles and the construction of localized Wannier functions (English)
[ Abstract ]
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.
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.
2016/11/29
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
Naotaka Shouji (Graduate School of Pure and Applied Sciences, University of Tsukuba)
Interior transmission eigenvalue problems on manifolds (Japanese)
Naotaka Shouji (Graduate School of Pure and Applied Sciences, University of Tsukuba)
Interior transmission eigenvalue problems on manifolds (Japanese)
2016/10/25
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
Hideshi YAMANE (Department of Mathematical Sciences, School of Science, Kwansei Gakuin University)
Asymptotics for the integrable discrete nonlinear Schr\"odinger equation (JAPANESE)
Hideshi YAMANE (Department of Mathematical Sciences, School of Science, Kwansei Gakuin University)
Asymptotics for the integrable discrete nonlinear Schr\"odinger equation (JAPANESE)
2016/07/12
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
X. P. Wang (Université de Nantes, France)
Gevrey estimates of the resolvent and sub-exponential time-decay (English)
X. P. Wang (Université de Nantes, France)
Gevrey estimates of the resolvent and sub-exponential time-decay (English)
[ Abstract ]
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.
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.
2016/06/28
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
Georgi Raikov (The Pontificia Universidad Católica de Chile)
Discrete spectrum of Schr\"odinger operators with oscillating decaying potentials (English)
Georgi Raikov (The Pontificia Universidad Católica de Chile)
Discrete spectrum of Schr\"odinger operators with oscillating decaying potentials (English)
[ Abstract ]
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.
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.
2016/06/21
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
Masao HIROKAWA (Institute of Engineering, Hiroshima University)
On the Hepp-Lieb-Preparata quantum phase transition for the quantum Rabi model (Japanese)
Masao HIROKAWA (Institute of Engineering, Hiroshima University)
On the Hepp-Lieb-Preparata quantum phase transition for the quantum Rabi model (Japanese)
[ Abstract ]
In my talk, I would like to consider the quantum Rabi model from the point of the view of quantum phase transition. Preparata claims that the ground state of the matter coupled with light is dressed with some photons provided that the coupling strength grows large though those photons should primarily be emitted from the matter in the relaxation of quantum states, and then, that the perturbative ground state switches with a non-perturbative one (Hepp-Lieb-Preparata quantum phase transition). He finds this based on the mathematical structure of the Hepp-Lieb quantum phase transition. Yoshihara and others recently showed an experimental result for the quantum Rabi model in circuit QED. In the experiment, they succeeded in demonstrating the so-called deep-strong coupling regime and the ground state dressed with a photon. We consider the quantum Rabi model in the light of the Hepp-Lieb-Preparata quantum phase transition. Our research is among the studies with aspects of mathematical physics, and deals with the A2-term problem.
In my talk, I would like to consider the quantum Rabi model from the point of the view of quantum phase transition. Preparata claims that the ground state of the matter coupled with light is dressed with some photons provided that the coupling strength grows large though those photons should primarily be emitted from the matter in the relaxation of quantum states, and then, that the perturbative ground state switches with a non-perturbative one (Hepp-Lieb-Preparata quantum phase transition). He finds this based on the mathematical structure of the Hepp-Lieb quantum phase transition. Yoshihara and others recently showed an experimental result for the quantum Rabi model in circuit QED. In the experiment, they succeeded in demonstrating the so-called deep-strong coupling regime and the ground state dressed with a photon. We consider the quantum Rabi model in the light of the Hepp-Lieb-Preparata quantum phase transition. Our research is among the studies with aspects of mathematical physics, and deals with the A2-term problem.
2016/06/14
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
NIIKUNI, Hiroaki (Maebashi Institute of Technology)
Schr¥"odinger operators on a periodically broken zigzag carbon nanotube (Japanese)
NIIKUNI, Hiroaki (Maebashi Institute of Technology)
Schr¥"odinger operators on a periodically broken zigzag carbon nanotube (Japanese)
2016/04/26
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
Saiei-Jaeyeong Matsubara-Heo (Graduate School of Mathematical Sciences, the University of Tokyo)
On microlocal analysis of Gauss-Manin connections for boundary singularities (Japanese)
Saiei-Jaeyeong Matsubara-Heo (Graduate School of Mathematical Sciences, the University of Tokyo)
On microlocal analysis of Gauss-Manin connections for boundary singularities (Japanese)
2016/04/12
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
Jussi Behrndt (Graz University of Technology)
Scattering matrices and Dirichlet-to-Neumann maps (English)
Jussi Behrndt (Graz University of Technology)
Scattering matrices and Dirichlet-to-Neumann maps (English)
[ Abstract ]
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.
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.
