An adiabatic theorem for section determinants of spectral projections

2005 ◽  
Vol 278 (4) ◽  
pp. 470-484 ◽  
Author(s):  
Peter Otte
Keyword(s):  
Adiabatic Theorem ◽  
Spectral Projections

scholarly journals A STRONG OPERATOR TOPOLOGY ADIABATIC THEOREM

2002 ◽  
Vol 14 (06) ◽  
pp. 569-584 ◽  
Author(s):  
ALEXANDER ELGART ◽  
JEFFREY H. SCHENKER
Keyword(s):  
Spectral Data ◽  
Time Scale ◽  
Strong Operator Topology ◽  
Continuous Functions ◽  
Adiabatic Theorem ◽  
Strong Operator ◽  
Intrinsic Time ◽  
Embedded Eigenvalues ◽  
Additive Perturbation ◽  
Spectral Projections

We prove an adiabatic theorem for the evolution of spectral data under a weak additive perturbation in the context of a system without an intrinsic time scale. For continuous functions of the unperturbed Hamiltonian the convergence is in norm while for a larger class functions, including the spectral projections associated to embedded eigenvalues, the convergence is in the strong operator topology.

scholarly journals Persistence of the spectral gap for the Landau–Pekar equations

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
Dario Feliciangeli ◽  
Simone Rademacher ◽  
Robert Seiringer
Keyword(s):  
Initial Data ◽  
Spectral Gap ◽  
Effective Hamiltonian ◽  
Adiabatic Theorem ◽  
Strongly Coupled

AbstractThe Landau–Pekar equations describe the dynamics of a strongly coupled polaron. Here, we provide a class of initial data for which the associated effective Hamiltonian has a uniform spectral gap for all times. For such initial data, this allows us to extend the results on the adiabatic theorem for the Landau–Pekar equations and their derivation from the Fröhlich model obtained in previous works to larger times.

scholarly journals Time-dependent quantum harmonic oscillator: a continuous route from adiabatic to sudden changes

2021 ◽  
Author(s):  
Daniel M. Tibaduiza ◽  
Luis Barbosa Pires ◽  
Carlos Farina
Keyword(s):  
Harmonic Oscillator ◽  
Simple Expression ◽  
Time Dependent ◽  
Frequency Change ◽  
Quantum Harmonic Oscillator ◽  
Transition Analysis ◽  
Adiabatic Theorem ◽  
Significant Difference ◽  
The One ◽  
Proper Interpretation

Abstract In this work, we give a quantitative answer to the question: how sudden or how adiabatic is a frequency change in a quantum harmonic oscillator (HO)? We do that by studying the time evolution of a HO which is initially in its fundamental state and whose time-dependent frequency is controlled by a parameter (denoted by ε) that can continuously tune from a totally slow process to a completely abrupt one. We extend a solution based on algebraic methods introduced recently in the literature that is very suited for numerical implementations, from the basis that diagonalizes the initial hamiltonian to the one that diagonalizes the instantaneous hamiltonian. Our results are in agreement with the adiabatic theorem and the comparison of the descriptions using the different bases together with the proper interpretation of this theorem allows us to clarify a common inaccuracy present in the literature. More importantly, we obtain a simple expression that relates squeezing to the transition rate and the initial and final frequencies, from which we calculate the adiabatic limit of the transition. Analysis of these results reveals a significant difference in squeezing production between enhancing or diminishing the frequency of a HO in a non-sudden way.

scholarly journals Spectral projections for the twisted Laplacian

2007 ◽  
Vol 180 (2) ◽  
pp. 103-110 ◽  
Author(s):  
Herbert Koch ◽  
Fulvio Ricci
Keyword(s):  
Twisted Laplacian ◽  
Spectral Projections
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