Fourier Series, Eigenvalue Problems, and Green's Function

2009 ◽  
pp. 403-452
Keyword(s):  
Fourier Series ◽  
Green’S Function ◽  
Green's Function ◽  
Eigenvalue Problems

scholarly journals GREEN'S FUNCTION OF A FINITE CHAIN AND THE DISCRETE FOURIER TRANSFORM

2006 ◽  
Vol 20 (05) ◽  
pp. 593-605 ◽  
Author(s):  
SERGIU COJOCARU
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Green’S Function ◽  
Discrete Fourier Transform ◽  
Series Expansion ◽  
Green's Function ◽  
Basis Set ◽  
Fourier Series Expansion ◽  
Open Boundary ◽  
Open Boundary Conditions

Unlike the Fourier series expansion, the discrete Fourier transform is defined on a finite basis set of harmonic functions. The first approach is widely used in condensed matter to describe the thermodynamic limit of various lattice models, while the latter did not receive sufficient development that would allow to address finite lattices. In the present paper a general expression for the Green's function of a finite one-dimensional lattice with nearest neighbor interaction is derived for the first time via discrete Fourier transform. Solution of the Heisenberg spin chain with periodic and open boundary conditions is considered as an example. Although the final expressions are completely equivalent to Bethe ansatz, the examples allow us to clarify the differences between the two approaches. On the other hand, it is explained why the well known results obtained by Fourier series expansion were incomplete and thus provides a deeper understanding of the approach.

A simplified method for integrating over Feynman histories

1957 ◽  
Vol 53 (3) ◽  
pp. 651-653 ◽  
Author(s):  
S. G. Brush
Keyword(s):  
Quantum Mechanics ◽  
Fourier Series ◽  
Schrödinger Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Schrodinger Equation ◽  
Space Time ◽  
Simplified Method ◽  
Time Formulation

ABSTRACTIn Feynman's ‘space-time’ formulation of quantum mechanics the Green's function for the Schrödinger equation is defined by an integral over all histories of the system. By integrating over one-parameter sets of functions, one gets the same Green's function as by integrating over a Fourier series, in simple cases. The method may be useful for estimating the result in cases when the integration over all histories cannot be performed exactly.

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