We address the problem of computing transport observables and spectral characteristics of quantum dynamics on arbitrary surfaces. Our approach is based on discrete exterior calculus (DEC) and applies to both open and closed quantum systems. We present an efficient algorithm for the calculation of the recursive Green’s functions (for open systems) and the full set of eigenfunctions and eigenvalues (for closed systems) using numerical tools available for DEC. Our approach is applied to the calculation of the conductance of a non-flat quantum device coupled to electron reservoirs and to obtain the spectra of ballistic cavities defined on curved surfaces. In both cases we found numerical evidences of a curvature induced integrable-chaotic crossover.

We introduce an integral transform that maps differential equations and special functions of standard continuous calculus onto finite difference equations and deformed special functions of mimetic discrete calculus, or h-calculus. We show that our procedure leads to insightful reformulations of several problems in mathematics and physics where discrete equations play a significant role, such as in solving finite difference equations, in applying discrete versions of integral transforms, such as the $h$-Laplace transform, in solving master equations of stochastic physics, in developing a discrete version of H theory of multiscale complex hierarchical phenomena and in finding lattice Green's functions for describing quantum charge transport through phase coherent systems. We believe that our integral transform technique, or mimetic map, will help systematize the connections through analogy between discrete calculus and standard continuous calculus.