scholarly journals Non-ergodic transformations with discrete spectrum

1965 ◽  
Vol 9 (2) ◽  
pp. 307-320 ◽  
Author(s):  
J. R. Choksi
Keyword(s):  
Discrete Spectrum ◽  
Ergodic Transformations

Ergodic transformations conjugate to their inverses by involutions

1996 ◽  
Vol 16 (1) ◽  
pp. 97-124 ◽  
Author(s):  
Geoffrey R. Goodson ◽  
Andrés del Junco ◽  
Mariusz Lemańczyk ◽  
Daniel J. Rudolph
Keyword(s):  
Discrete Spectrum ◽  
Probability Space ◽  
Closure Property ◽  
Simple Spectrum ◽  
Ergodic Transformations ◽  
Ergodic Automorphism ◽  
Singular Spectrum ◽  
Weak Closure ◽  
Identity Automorphism ◽  
Borel Probability

AbstractLetTbe an ergodic automorphism defined on a standard Borel probability space for whichTandT−1are isomorphic. We investigate the form of the conjugating automorphism. It is well known that ifTis ergodic having a discrete spectrum andSis the conjugation betweenTandT−1, i.e.SsatisfiesTS=ST−1thenS2=Ithe identity automorphism. We show that this result remains true under the weaker assumption thatThas a simple spectrum. IfThas the weak closure property and is isomorphic to its inverse, it is shown that the conjugationSsatisfiesS4=I. Finally, we construct an example to show that the conjugation need not be an involution in this case. The example we construct, in addition to having the weak closure property, is of rank two, rigid and simple for all orders with a singular spectrum of multiplicity equal to two.

scholarly journals Transformations With Discrete Spectrum are Stacking Transformations

1976 ◽  
Vol 28 (4) ◽  
pp. 836-839 ◽  
Author(s):  
Andrés Del Junco
Keyword(s):  
Discrete Spectrum ◽  
Ergodic Theory ◽  
Great Success ◽  
Ergodic Transformations ◽  
Simple Characterization

The stacking method (see [1] and [5, Section 6]) has been used with great success in ergodic theory to construct a wide variety of examples of ergodic transformations (see, for example, [1 ; 3 ; 4; 5; 7]). However very little is known in general about the class of transformations which can be constructed by the stacking method using single stacks. In particular there is no simple characterization of the class .

Some Aspects of Discrete Relaxation Time Spectra as Obtained from Dynamic Moduli Data

1995 ◽  
Vol 60 (11) ◽  
pp. 1815-1829 ◽  
Author(s):  
Jaromír Jakeš
Keyword(s):  
Relaxation Time ◽  
Least Squares ◽  
Discrete Spectrum ◽  
Integral Transform ◽  
Time Spectrum ◽  
Relaxation Time Spectrum ◽  
Dynamic Moduli ◽  
Best Fitting ◽  
Discrete Spectra ◽  
Well Posed

The problem of finding a relaxation time spectrum best fitting dynamic moduli data in the least-squares sense is shown to be well-posed and to yield a discrete spectrum, provided the data cannot be fitted exactly, i.e., without any deviation of data and calculated values. Properties of the resulting spectrum are discussed. Examples of discrete spectra obtained from simulated literature data and experimental literature data on polymers are given. The problem of smoothing discrete spectra when continuous ones are expected is discussed. A detailed study of an integral transform inversion under the non-negativity constraint is given in Appendix.

Localization of Discrete Spectrum of Multiparticle Schrödinger Operators

1985 ◽  
Vol 40 (10) ◽  
pp. 1052-1058 ◽  
Author(s):  
Heinz K. H. Siedentop
Keyword(s):  
Discrete Spectrum ◽  
Lower Bounds ◽  
Upper Bound ◽  
Schrödinger Operators ◽  
Upper And Lower Bounds ◽  
Schrodinger Operators

An upper bound on the dimension of eigenspaces of multiparticle Schrödinger operators is given. Its relation to upper and lower bounds on the eigenvalues is discussed.

scholarly journals Eigenvalue bounds for non-selfadjoint Dirac operators

2021 ◽  
Author(s):  
Piero D’Ancona ◽  
Luca Fanelli ◽  
Nico Michele Schiavone
Keyword(s):  
Dirac Operator ◽  
Discrete Spectrum ◽  
Complex Plane ◽  
Dirac Operators ◽  
Resolvent Estimates ◽  
Eigenvalue Bounds ◽  
Mixed Norms ◽  
Massless Case ◽  
Abstract Version ◽  
Embedded Eigenvalues

AbstractWe prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

BOSE–EINSTEIN STATISTICS IN A DISCRETE SPECTRUM TRAP

2004 ◽  
Vol 18 (04n05) ◽  
pp. 555-563 ◽  
Author(s):  
ENRICO CELEGHINI ◽  
MARIO RASETTI
Keyword(s):  
Critical Temperature ◽  
Low Temperature ◽  
Specific Heat ◽  
Discrete Spectrum ◽  
Harmonic Trap ◽  
Statistical Properties ◽  
Bose Einstein ◽  
The Continuum

A detailed description of the statistical properties of a system of bosons in a harmonic trap at low temperature, which is expected to bear on the process of BE condensation, is given resorting only to the basic postulates of Gibbs and Bose, without assuming equipartition nor continuum statistics. Below Tc such discrete spectrum theory predicts for the thermo-dynamical variables a behavior different from the continuum case. In particular a new critical temperature Td emerges where the specific heat exhibits a λ-like spike.

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