Abstract In order to describe quantum mechanical effects, the use of von-Neumann equation is apparent. this work, we present a unified numerical framework so that in center-of-mass coordinates leads Quantum Liouville-type when choosing suitable basis. particular, proposed approach can be related conventional Wigner plane wave basis used. The drawback methods high computational cost. Our present...

In this paper we establish a Liouville type theorem for fully nonlinear elliptic equations related to a conjecture of De Giorgi in IR2. We prove that if the level lines of a solution have bounded curvature, then these level lines are straight lines. As a consequence, the solution is one-dimensional. The method also provides a result on free boundary problems of Serrin type.

We prove a Liouville-type theorem for bounded stable solutions v ∈ C(R) of elliptic equations of the type (−∆)v = f(v) in R, where s ∈ (0, 1) and f is any nonnegative function. The operator (−∆) stands for the fractional Laplacian, a pseudo-differential operator of symbol |ξ|.