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We study propagation of phase space singularities for the initial value Cauchy problem for a class of Schrodinger equations. The Hamiltonian is the Weyl quantization of a quadratic form whose real part is non-negative. The equations are studied in the framework of projective Gelfand--Shilov spaces and their distribution duals. The corresponding notion of singularities is called the Gelfand--Shilov wave front set and means the lack of exponential decay in open cones in phase space. Our main result shows that the propagation is determined by the singular space of the quadratic form, just as in the framework of the Schwartz space, where the notion of singularity is the Gabor wave front set.
We revisit the paper [Mel86] by R. Melrose, providing a full proof of the main theorem on propagation of singularities for subelliptic wave equations, and linking this result with sub-Riemannian geometry. This result asserts that singularities of sub
We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $ipartial_t u + (-Delta)^{frac{alpha}2} u = 2gammabeta e^{beta|u|^2}u$ on $d$-dimensional compact Riemannian manifolds $mathcal{M}
This paper deals with the 2-D Schrodinger equation with time-oscillating exponential nonlinearity $ipartial_t u+Delta u= theta(omega t)big(e^{4pi|u|^2}-1big)$, where $theta$ is a periodic $C^1$-function. We prove that for a class of initial data $u_0
We study a nonlinear equation in the half-space ${x_1>0}$ with a Hardy potential, specifically [-Delta u -frac{mu}{x_1^2}u+u^p=0quadtext{in}quad mathbb R^n_+,] where $p>1$ and $-infty<mu<1/4$. The admissible boundary behavior of the positive solution
The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation begin{equation}label{eq:discount 1}tag{HJ$_lambda$} lambda v(x)+H( x, Dv(x) )=0 , quad xin mathbb{R}^n.