Abstract
The fundamental solutionE(t,s,x,y) of time dependent Schrödinger equationsi∂u/∂t=−(1/2)Δu+V(t,x)u is studied. It is shown that
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•E(t,s,x,y) is smooth and bounded fort≠s if the potential is sub-quadratic in the sense thatV(t,x)=o(|x|2) at infinity;
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• in one dimension, ifV(t,x)=V(x) is time independent and super-quadratic in the sense thatV(x)≧C(1+|x|)2+ε at infinity,C>0 and ε>0, thenE(t,s,x,y) is nowhereC 1.
The result is explained in terms of the limiting behavior as the energy tends to infinity of the corresponding classical particle.
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Yajima, K. Smoothness and non-smoothness of the fundamental solution of time dependent Schrödinger equations. Commun.Math. Phys. 181, 605–629 (1996). https://doi.org/10.1007/BF02101289
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DOI: https://doi.org/10.1007/BF02101289