scholarly journals Pure discrete spectrum for a class of one-dimensional substitution tiling systems

2015 ◽  
Vol 36 (3) ◽  
pp. 1159-1173 ◽  
Author(s):  
Marcy Barge
Keyword(s):  
Discrete Spectrum ◽  
One Dimensional ◽  
Substitution Tiling ◽  
Tiling Systems

scholarly journals TIME-DEPENDENT PARASUPERSYMMETRY IN QUANTUM MECHANICS

1996 ◽  
Vol 11 (26) ◽  
pp. 2095-2104 ◽  
Author(s):  
BORIS F. SAMSONOV
Keyword(s):  
Quantum Mechanics ◽  
Discrete Spectrum ◽  
Schrodinger Equation ◽  
Free Particle ◽  
Time Dependent ◽  
Darboux Transformations ◽  
Integral Of Motion ◽  
One Dimensional ◽  
A Chain ◽  
The One

Parasupersymmetry of the one-dimensional time-dependent Schrödinger equation is established. It is intimately connected with a chain of the time-dependent Darboux transformations. As an example a parasupersymmetric model of nonrelativistic free particle with threefold degenerate discrete spectrum of an integral of motion is constructed.

scholarly journals ON THE SPECTRUM OF THE ONE-DIMENSIONAL SCHRÖDINGER HAMILTONIAN PERTURBED BY AN ATTRACTIVE GAUSSIAN POTENTIAL

2017 ◽  
Vol 57 (6) ◽  
pp. 385 ◽  
Author(s):  
Silvestro Fassari ◽  
Manuel Gadella ◽  
Luis Miguel Nieto ◽  
Fabio Rinaldi
Keyword(s):  
Discrete Spectrum ◽  
Energy Levels ◽  
Trace Class ◽  
Bound State ◽  
Great Accuracy ◽  
Coupling Constant ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Gaussian Potential ◽  
The One

<p>We propose a new approach to the problem of finding the eigenvalues (energy levels) in the discrete spectrum of the one-dimensional Hamiltonian with an attractive Gaussian potential by using the well-known Birman-Schwinger technique. However, in place of the Birman-Schwinger integral operator we consider an isospectral operator in momentum space, taking advantage of the unique feature of this potential, that is to say its invariance under Fourier transform. <br />Given that such integral operators are trace class, it is possible to determine the energy levels in the discrete spectrum of the Hamiltonian as functions of <span>the coupling constant with great accuracy by solving a finite number of transcendental equations. We also address the important issue of the coupling constant thresholds of the Hamiltonian, that is to say the critical values of λ for which we have the emergence of an additional bound state out of the absolutely continuous spectrum. </span></p>

scholarly journals Beyond primitivity for one-dimensional substitution subshifts and tiling spaces

2016 ◽  
Vol 38 (3) ◽  
pp. 1086-1117 ◽  
Author(s):  
GREGORY R. MALONEY ◽  
DAN RUST
Keyword(s):  
Invariant Subspaces ◽  
Finite Graph ◽  
One Dimension ◽  
One Dimensional ◽  
Substitution Tiling ◽  
Cohomological Invariants ◽  
Tiling Spaces ◽  
Tiling Space ◽  
Cech Cohomology

We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterization of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution. We then extend to the non-minimal setting a result obtained by Anderson and Putnam for primitive substitutions, which says that a substitution tiling space is homeomorphic to an inverse limit of a certain finite graph under a self-map induced by the substitution. We use this result to explore the structure of the lattice of closed invariant subspaces and quotients of a substitution tiling space, for which we compute cohomological invariants that are stronger than the Čech cohomology of the tiling space alone.

Primitive Tilings and Coherent Frames

2021 ◽  
Vol 60 ◽  
pp. 47-64
Author(s):  
Luis Silvestre ◽  
◽  
Eden Miro ◽  
Job Nable
Keyword(s):  
Lie Groups ◽  
Primitive Substitution ◽  
Substitution Tiling ◽  
Tiling Systems ◽  
Representation Spaces ◽  
Tiling Space

This paper characterizes primitive substitution tiling systems for Lie groups for which every element of the associated tiling space generates coherent frames for corresponding representation spaces.

scholarly journals Cohomology of substitution tiling spaces

2009 ◽  
Vol 30 (6) ◽  
pp. 1607-1627 ◽  
Author(s):  
MARCY BARGE ◽  
BEVERLY DIAMOND ◽  
JOHN HUNTON ◽  
LORENZO SADUN
Keyword(s):  
Inverse Limit ◽  
Direct Limit ◽  
Higher Dimensions ◽  
Rotation Group ◽  
One Dimensional ◽  
Substitution Tiling ◽  
Tiling Spaces ◽  
Tiling Space ◽  
Cellular Complex

AbstractAnderson and Putnam showed that the cohomology of a substitution tiling space may be computed by collaring tiles to obtain a substitution which ‘forces its border’. One can then represent the tiling space as an inverse limit of an inflation and substitution map on a cellular complex formed from the collared tiles; the cohomology of the tiling space is computed as the direct limit of the homomorphism induced by inflation and substitution on the cohomology of the complex. In earlier work, Barge and Diamond described a modification of the Anderson–Putnam complex on collared tiles for one-dimensional substitution tiling spaces that allows for easier computation and provides a means of identifying certain special features of the tiling space with particular elements of the cohomology. In this paper, we extend this modified construction to higher dimensions. We also examine the action of the rotation group on cohomology and compute the cohomology of the pinwheel tiling space.

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