Differences

This shows you the differences between two versions of the page.

--- cwf:space-analyticity_of_wave_function:coulomb_potential [2016/11/25 19:07] – markus
+++ cwf:space-analyticity_of_wave_function:coulomb_potential [2017/01/24 14:42] (current) – markus
@@ Line 7: / Line 7: @@
 > On the other hand, if $V$ is not smooth, e.g., if $V$ is the Coulomb potential in dimension three, the singularities of $V$ create those of the FDS and $E(t, x, y)$ [the fundamental solution] is not smooth everywhere. However, the strong dissipation property of the free propagator $\mathrm{e}^{-\mathrm{itH_0}}$ moderates the singularities and we expect that $E(t, x, y)$ is bounded and continuous for $t\neq 0$ if $V$ is bounded at infinity in a suitable norm and is not too singular locally (see Simon [14] [ref. [(:cite:cwf:simon1982)]] who conjectures that this is true if $V$ is of Kato class).
-The author goes on and shows boundedness and continuity for the fundamental solution for a class that includes the Coulomb potential in dimension three. Finally one should note that for a Hamiltonian with Coulomb potentials and even including Coulomb interactions between multiple particles, it is known that the eigenstate is analytic away from the potential sigularities and has $|x|$-formed cusps at the location of the singularities [(:cite:cwf:fournais2009analytic)].
+The author goes on and shows boundedness and continuity for the fundamental solution for a class that includes the Coulomb potential in dimension three. Finally one should note that for a Hamiltonian with Coulomb potentials and even including Coulomb interactions between multiple particles, it is known that an eigenstate is analytic away from the potential sigularities and has $|x|$-formed cusps at the location of the singularities [(:cite:cwf:fournais2009analytic)].