Very Weak Solution of the Discrete Wave Equation for Harmonic Oscillator

In this paper, we announce some preliminary findings of the joint work with [6] for the wave equation with Schrödinger operator. In particular, we study the semiclassical analogue of the wave equation for the discrete harmonic oscillator on the lattice \(\hbar \mathbb {Z}^{n}\) . In the case of regular coefficients, we show that the Cauchy problem is well-posed in the associated Sobolev-type spaces. Moreover, we also show that the Cauchy problem is very weakly well-posed when the coefficients are allowed to have a distributional singularity.