Semi-classical Pseudo-differential Operators on $$\hbar \mathbb {Z}^n$$ and Applications

In this paper we consider the semiclassical version of pseudo-differential operators on the lattice space \(\hbar {{\mathbb {Z}}^{n}}\). The current work is an extension of the previous work (Botchway et al. in J Funct Anal 278(11):108473, 33, 2020) and agrees with it in the limit of the parameter \(\hbar \rightarrow 1\). The various representations of the operators will be studied as well as the composition, transpose, adjoint and the link between ellipticity and parametrix of operators. We also give the conditions for the \(\ell ^p\), weighted \(\ell ^2\) boundedness and \(\ell ^p\) compactness of operators. We investigate the relation between the classical and semi-classical quantization in the spirit of Ruzhansky and Turunen (Pseudo-differential operators and symmetries. Pseudo-differential operators, vol 2. Theory and Applications, Birkhäuser, Basel, 2010; J Fourier Anal Appl 16(6):943–982, 2010) RTspsJFAA and employ its applications to Schatten–von Neumann classes on \(\ell ^2( \hbar \mathbb {Z}^n)\). We establish Gårding and sharp Gårding inequalities, with an application to the well-posedness of parabolic equations on the lattice \(\hbar \mathbb {Z}^n\). Finally we verify that in the limiting case where \(\hbar \rightarrow 0\) the semi-classical calculus of pseudo-differential operators recovers the classical Euclidean calculus, but with a twist.