Continuous sums of measures and continuous spectra

Von Neumann's definition of the continuous sum of Hilbert spaces led Segal [3] to define the continuous sum of measures on a measurable space. In this note we employ Segal's definition to investigate the measure structures associated with a self-adjoint transformation of pure point spectrum and a self-adjoint transformation of pure continuous spectrum. While these transformations, as operators on separable Hilbert spaces, are the antithesis of each other we show that in their measure structure one is a particular case of the other.

Download Full-text

Cut-and-Project Schemes for Pisot Family Substitution Tilings

Symmetry ◽

10.3390/sym10100511 ◽

2018 ◽

Vol 10 (10) ◽

pp. 511 ◽

Cited By ~ 1

Author(s):

Jeong-Yup Lee ◽

Shigeki Akiyama ◽

Yasushi Nagai

Keyword(s):

Point Spectrum ◽

The Other ◽

Pure Point ◽

Pure Point Spectrum ◽

Internal Space ◽

Sufficient Condition ◽

Abstract Space ◽

Dynamical Spectrum ◽

Substitution Tiling ◽

Substitution Tilings

We consider Pisot family substitution tilings in R d whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space R m for some integer m ∈ N defined from the Pisot family property, and the second CPS has an internal space H that is an abstract space defined from the condition of the pure point spectrum. However, it is not known how these two CPSs are related. Here we provide a sufficient condition to make a connection between the two CPSs. For Pisot unimodular substitution tiling in R , the two CPSs turn out to be same due to the remark by Barge-Kwapisz.

Download Full-text

Spectral Theory for the Neumann Laplacian on Planar Domains With Horn-Like Ends

Canadian Journal of Mathematics ◽

10.4153/cjm-1997-012-8 ◽

1997 ◽

Vol 49 (2) ◽

pp. 232-262 ◽

Cited By ~ 2

Author(s):

Julian Edward

Keyword(s):

Large Class ◽

Continuous Spectrum ◽

Spectral Theory ◽

Point Spectrum ◽

Pure Point ◽

Pure Point Spectrum ◽

Singular Continuous Spectrum ◽

Finite Multiplicity ◽

Planar Domains ◽

Neumann Laplacian

AbstractThe spectral theory for the Neumann Laplacian on planar domains with symmetric, horn-like ends is studied. For a large class of such domains, it is proven that the Neumann Laplacian has no singular continuous spectrum, and that the pure point spectrum consists of eigenvalues of finite multiplicity which can accumulate only at 0 or ∞. The proof uses Mourre theory.

Download Full-text

Cut-and-Project Schemes for Pisot Family Substitution Tilings

10.20944/preprints201809.0366.v1 ◽

2018 ◽

Author(s):

Jeong-Yup Lee ◽

Shigeki Akiyama ◽

Yasushi Nagai

Keyword(s):

Point Spectrum ◽

The Other ◽

Pure Point ◽

Pure Point Spectrum ◽

Internal Space ◽

Sufficient Condition ◽

Abstract Space ◽

Dynamical Spectrum ◽

Substitution Tiling ◽

Substitution Tilings

We consider Pisot family substitution tilings in $\R^d$ whose dynamical spectrum is pure point. There are two cut-and-project schemes(CPS) which arise naturally: one from the Pisot family property and the other from the pure point spectrum respectively. The first CPS has an internal space $\R^m$ for some integer $m \in \N$ defined from the Pisot family property, and the second CPS has an internal space $H$ which is an abstract space defined from the property of the pure point spectrum. However it is not known how these two CPS's are related. Here we provide a sufficient condition to make a connection between the two CPS's. In the case of Pisot unimodular substitution tiling in $\R$, the two CPS's turn out to be same due to [5, Remark 18.5].

Download Full-text