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Infinite Analysis Seminar Tokyo
Seminar information archive ～04/19｜Next seminar｜Future seminars 04/20～
| Date, time & place | Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
|---|
2014/12/11
15:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Yohei Kashima (Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30
Renormalization group method for many-electron systems (JAPANESE)
Unitary transformations and multivariate special
orthogonal polynomials (JAPANESE)
Yohei Kashima (Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30
Renormalization group method for many-electron systems (JAPANESE)
[ Abstract ]
We consider quantum many-body systems of electrons
hopping and interacting on a lattice at positive temperature.
As it is possible to write down each order term rigorously in
principle, the perturbation series expansion with the coupling
constant between electrons is thought as a valid method to
compute physical quantities. By directly estimating each term,
one can prove that the perturbation series is convergent if the
coupling constant is less than some power of temperature. This is,
however, a serious constraint for models of interacting electrons
in low temperature. In order to ensure the analyticity of physical
quantities of many-electron systems with the coupling constant in
low temperature, renormalization group methods have been developed
in recent years. As one progress in this direction, we construct a
renormalization group method for the half-filled Hubbard model on a
square lattice, which is a typical model of many-electron, and prove
the following. If the system contains the magnetic flux pi (mod 2 pi)
per plaquette, the free energy density of the system is analytic with
the coupling constant in a neighborhood of the origin and it uniformly
converges to the infinite-volume, zero-temperature limit. It is known
that the flux pi condition is sufficient for the free energy density
to be minimum. Thus, it follows that the same analyticity and the
convergent property hold for the minimum free energy density of the
system.
Genki Shibukawa (Institute of Mathematics for Industory, Kyushu University) 17:00-18:30We consider quantum many-body systems of electrons
hopping and interacting on a lattice at positive temperature.
As it is possible to write down each order term rigorously in
principle, the perturbation series expansion with the coupling
constant between electrons is thought as a valid method to
compute physical quantities. By directly estimating each term,
one can prove that the perturbation series is convergent if the
coupling constant is less than some power of temperature. This is,
however, a serious constraint for models of interacting electrons
in low temperature. In order to ensure the analyticity of physical
quantities of many-electron systems with the coupling constant in
low temperature, renormalization group methods have been developed
in recent years. As one progress in this direction, we construct a
renormalization group method for the half-filled Hubbard model on a
square lattice, which is a typical model of many-electron, and prove
the following. If the system contains the magnetic flux pi (mod 2 pi)
per plaquette, the free energy density of the system is analytic with
the coupling constant in a neighborhood of the origin and it uniformly
converges to the infinite-volume, zero-temperature limit. It is known
that the flux pi condition is sufficient for the free energy density
to be minimum. Thus, it follows that the same analyticity and the
convergent property hold for the minimum free energy density of the
system.
Unitary transformations and multivariate special
orthogonal polynomials (JAPANESE)
[ Abstract ]
Investigations into special orthogonal function systems by
using unitary transformations have a long history.
This is, by calculating an image of some unitary transform (e.g. Fourier
trans.) of a known orthogonal system, we derive a new orthogonal system
and obtain its fundamental properties.
This basic concept and technique have been known since ancient times for
a single variable case, and recently these multivariate analogue has
been studied by Davidson, Olafsson, Zhang, Faraut, Wakayama et.al..
In our talk, we introduce the unitary picture for the circular Jacobi
polynomials obtained by Shen, further give a multivariate analogue of
the results of Shen.
These polynomials, which we call multivariate circular Jacobi (MCJ)
polynomials, are generalizations (2-parameter deformation) of the
spherical (zonal) polynomials that are different from the Jack or
Macdonald polynomials, which are well known as an extension of spherical
polynomials.
We also remark that the weight function of their orthogonality relation
coincides with the circular Jacobi ensemble defined by Bourgade,et,al.,
and the modified Cayley transform of the MCJ polynomials satisfy with
some quasi differential equation.
In addition, we can give a generalization of MCJ polynomials as
including the Jack polynomials.
For this generalized MCJ polynomials, we would like to present some
conjectures and problems.
If we have time, we also describe a unitary picture of Meixner,
Charlier and Krawtchouk polynomials which are typical examples of
discrete orthogonal systems, and mention their multivariate analogue.
Investigations into special orthogonal function systems by
using unitary transformations have a long history.
This is, by calculating an image of some unitary transform (e.g. Fourier
trans.) of a known orthogonal system, we derive a new orthogonal system
and obtain its fundamental properties.
This basic concept and technique have been known since ancient times for
a single variable case, and recently these multivariate analogue has
been studied by Davidson, Olafsson, Zhang, Faraut, Wakayama et.al..
In our talk, we introduce the unitary picture for the circular Jacobi
polynomials obtained by Shen, further give a multivariate analogue of
the results of Shen.
These polynomials, which we call multivariate circular Jacobi (MCJ)
polynomials, are generalizations (2-parameter deformation) of the
spherical (zonal) polynomials that are different from the Jack or
Macdonald polynomials, which are well known as an extension of spherical
polynomials.
We also remark that the weight function of their orthogonality relation
coincides with the circular Jacobi ensemble defined by Bourgade,et,al.,
and the modified Cayley transform of the MCJ polynomials satisfy with
some quasi differential equation.
In addition, we can give a generalization of MCJ polynomials as
including the Jack polynomials.
For this generalized MCJ polynomials, we would like to present some
conjectures and problems.
If we have time, we also describe a unitary picture of Meixner,
Charlier and Krawtchouk polynomials which are typical examples of
discrete orthogonal systems, and mention their multivariate analogue.
