Abstract
We calculate in a numerically friendly way the Fourier transform of a non-integrable function, such as , by replacing F with R-1FR, where R represents the resolvent for harmonic oscillator Hamiltonian. As contrasted with the non-analyticity of at in the case of a simple replacement of F by , where and represent the momentum and position operators, respectively, the turns out to be an entire function. In calculating the resolvent kernel, the sampling theorem is of great use. The resolvent based Fourier transform can be made supersymmetric (SUSY), which not only makes manifest the usefulness of the even-odd decomposition ofin a more natural way, but also leads to a natural definition of SUSY Fourier transform through the commutativity with the SUSY resolvent.
Highlights
Fourier transform (FT) : L2 ( ) → L2 ( ) by ( φ )( x) = ∫ eikx φ (k)dk, which is a unitary operator, is a fundamental method in function analysis and is applied to many fields in physics
In calculating the resolvent kernel, the sampling theorem is of great use
The corresponding self-adjoint operator is given by the harmonic oscillator Hamiltonian : L2 ( ) → L2 ( ) by ( φ )( x=) ( ) 1 p 2 + q 2 −1 φ ( x), (1)
Summary
)dk, which is a unitary operator, is a fundamental method in function analysis and is applied to many fields in physics. The corresponding self-adjoint operator is given by the harmonic oscillator Hamiltonian : L2 ( ) → L2 ( ) by ( φ )( x=) ( ) 1 p 2 + q 2 −1 φ ( x), (1). (multipled by e−x2 2 ) is a simultaneous eigenfunction of and , with their eigenvalues given by in and n , respectively. If a function φ : → is integrable, its FT is well defined. If the function φ is not integrable, for example φ ( x) = 1, its FT should be regarded as a generalized function.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
