平成30年度学習院大学スペクトル理論セミナー




第167回学習院大学スペクトル理論セミナー

日時:2018 年 11月 12 日 (月) 16:30 -- 17:30
場所: 学習院大学 南4号館2階 205セミナー室

講演者: Claudio Cacciapuoti 氏 (University of Insburia)

Title:Scattering from local deformation of a semitransparent plane

Abstract: We discuss the scattering problem for a couple of Schrödinger operators in dimension three in a setting that resembles the presence of an unbounded obstacle. The reference operator in the scattering couple is the one formally defined as the Laplacian plus a delta-interaction supported by a plane. We consider a surface obtained through a local deformation of the plane, given by the graph of a compactly supported function, and define the perturbed operator as the Laplacian plus a delta-interaction supported by such surface. We show asymptotic completeness of the corresponding wave operators, provide a limiting absorption principle, and give a representation formula for the scattering matrix. Moreover, we show that as the deformation goes to zero the scattering matrix converges to the identity.
The talk is based on the joint work with Davide Fermi and Andrea Posilicano : arXiv:1807.07916 [math-ph]




第166回学習院大学スペクトル理論セミナー

日時:2018 年 7月 7 日 (土) 16:30 -- 17:30
場所: 学習院大学 南4号館2階 205セミナー室

講演者: Arne Jensen 氏 (Aalborg Univ.)

Title:On local extreme of the norm of the resolvent of an operator with compact resolvent

Abstract: We study the norm of the resolvent of a densely defined closed operator with compact resolvent on a Hilbert space. It is shown that for any point in the resolvent set there exist directions in which the norm grows at least quadratically with the distance from this point. As a consequence we obtain a new proof of the well known fact that the norm of the resolvent has no local maxima. Our proof does not use the maximum principle. It is based on a direct estimate of the norm using the Schur decomposition.
Joint work with H. Cornean, H. Garde, and H. K. Knörr (all at Aalborg Univ.)




第165回学習院大学スペクトル理論セミナー

日時:2018 年 6月 9 日 (土) 15:00 -- 17:30
場所: 学習院大学 南4号館2階 205セミナー室

1. 15:00--16:00
講演者: Khanh Duy Trinh氏 (東北大・理)

Title:Classical beta ensembles in global regime

Abstract: Gaussian beta ensembles, Wishart beta ensembles and Jacobi beta ensembles are three typical examples of beta ensembles on the real line. The limiting behavior of the empirical distribution, the probability measure putting equal mass at each eigenvalue, is one of fundamental problem in studying a random matrix model. Based on their random tridiagonal matrix models, this talk introduces a unify approach to completely solve that problem.The result can be shortly mentioned as follows. The limit of the empirical distribution depends only on the limit of the coupling \( n \beta \): zero, finite or infinite. Here \(n\) is the system size, or the order of a random matrix, and \( n \beta \) is a parameter regarded as the inverse temperature of the system. In particular, the case where \( n \beta \) tends to infinity is similar to the case of fixed beta. When \( n \beta \) tends to a finite constant, the limiting distribution is an associated version of the corresponding weight in the sense of orthogonal polynomials.

16:00--16:30 tea time

2. 16:30--17:30
講演者: 上木 直昌氏 (京都大・人間環境)

Title: Asymptotic behavior of the integrated density of states of a Schrödinger operator with positive potentials located around all sample points of some random point field

Abstract:Asymptotic behavior of the integrated density of states of a Schrödinger operator with positive potentials located around all sample points of some random point field at the infimum of the spectrum is investigated. The random point field is taken from a subclass of the class given by Shirai and Takahashi in terms of the Fredholm determinant. In the subclass, the obtained leading orders are same with the well known results for the Poisson point fields, and the character of the random field appears in the leading constants. The random point field associated with the sine kernel and the Ginibre random point field are well studied examples not included in the above subclass, though they are included in the class by Shirai and Takahashi. By applying the results on asymptotics of the hole probability for these random fields, the corresponding asymptotic behaviors of the densities of the states are also investigated in the case where the single site potentials have compact supports.




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