Considering the real-time quantum propagator originated by the green's function of the time-dependent Schrödinger equation:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]i\hbar \frac{d\psi}{dt} =H\psi[/itex]

namely,

[itex]\psi(t) =e^{-i\hbar H t}\psi(0) [/itex]

it is straightforward to see that using as basis set the eigenvector of

[itex]H\phi_k =\varepsilon_k\phi_k [/itex]

The matrix representation of the propagator is diagonal (maximaly sparse)

[itex]e^{-i\hbar H t}=\sum_{k}e^{-i\hbar\varepsilon_k}|\phi_k><\phi_k|.[/itex]

My question is: To which extent one may claim that the matrix representation of the propagator is sparse, if a basis set different from the eigenbasis (e.g. a overcomplete family of Hermite functions) is employed to represent the propagator considering that the potential in the Hamiltonian is an analytic function?

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# Concerning the sparcity of the real-time quantum propagator

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