The Heat Equation for the Hermite Operator on the Heisenberg Group∗

Translation invariant surfaces in the 3-dimensional Heisenberg‎ ‎group

‎In this paper‎, ‎we study translation invariant surfaces in the‎ ‎3-dimensional Heisenberg group $rm Nil_3$‎. ‎In particular‎, ‎we‎ ‎completely classify translation invariant surfaces in $rm Nil_3$‎ ‎whose position vector $x$ satisfies the equation $Delta x = Ax$‎, ‎where $Delta$ is the Laplacian operator of the surface and $A$‎ ‎is a $3 times 3$-real matrix‎.

φ-factorable operators and Weyl-Heisenberg frames on LCA groups

A Note on the Heat Kernel on the Heisenberg Group

Let ps be the convolution kernel of the operator e −sL (see [5, (1.10), (1.11)]). When s > 0, e−sL is the solution operator for the Heisenberg heat equation ∂su = −Lu and ps is called the heat kernel (see [6, (7.30), p. 71]. The goal of this note is to study the analytic continuation of the heat kernel ps. This is interesting from the point of view of the theory of analytic hypoellipticity (see...

Heisenberg Uncertainty Relation in Quantum Liouville Equation

We consider the quantum Liouville equation and give a characterization of the solutions which satisfy the Heisenberg uncertainty relation. We analyze three cases. Initially we consider a particular solution of the quantum Liouville equation: the Wigner transform f x,v,t of a generic solution ψ x;t of the Schrödinger equation. We give a representation of ψ x, t by the Hermite functions. We show ...

The Explicit Solution of the ∂̄-neumann Problem in a Non-isotropic Siegel Domain

In this paper, we solve the ∂̄-Neumann problem on (0, q) forms, 0 ≤ q ≤ n, in the strictly pseudoconvex non-isotropic Siegel domain: U = (z, zn+1) : z = (z1, · · · , zn) ∈ C , zn+1 ∈ C; =m(zn+1) > n ∑ j=1 aj |zj |  , where aj > 0 for j = 1, 2, · · · , n. The metric we use is invariant under the action of the Heisenberg group on the domain. The fundamental solution of the related differen...