Frozen Gaussian series representation of the imaginary time propagator theory and numerical tests

2009 ◽  
Vol 131 (4) ◽  
pp. 044116 ◽  
Author(s):  
Dong H. Zhang ◽  
Jiushu Shao ◽  
Eli Pollak
Keyword(s):  
Series Representation ◽  
Imaginary Time ◽  
Numerical Tests ◽  
Gaussian Series ◽  
Propagator Theory

The Laguerre-Gaussian series representation of two-dimensional fractional Fourier transform

1998 ◽  
Vol 31 (46) ◽  
pp. 9353-9357 ◽  
Author(s):  
Li Yu ◽  
Wenda Huang ◽  
Meichun Huang ◽  
Zizhong Zhu ◽  
Xiaoming Zeng ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Series Representation ◽  
Two Dimensional ◽  
Gaussian Series

Thermodynamic Green’s Functions and Spectral Structure

2018 ◽  
Author(s):  
Norman J. Morgenstern Horing
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Spectral Weight ◽  
Statistical Thermodynamic ◽  
Spectral Structure ◽  
Imaginary Time ◽  
Translational Invariance ◽  
The One

Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.

scholarly journals R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

2021 ◽  
pp. 1-61
Author(s):  
Fan Gao
Keyword(s):  
Series Representation ◽  
Principal Series ◽  
Symplectic Groups ◽  
Principal Series Representation ◽  
Connected Group ◽  
Covering Group ◽  
Simply Connected

Abstract For a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

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