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In this paper, we show discretized Lyapunov functions for various nonlinear dynamical systems by using the method that the authors have proposed. We use Kushner's scheme of difference approximation with directions and Alcaraz et al.'s quantization of Markov processes to approximate Lyapunov equations by linear Schroumldinger-like equations. We construct time-invariant functions concerned with the solutions to Schroumldinger-like equations. Then, we show the effectiveness of the method via an example.