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In this paper, we apply the solution of the Schrödinger equation, i.e. the Schrödinger operator, to the graph characterization problem. The motivation behind this approach is two-fold. Firstly, the mathematically similar heat kernel has been used in the past for this same problem. And secondly, due to the quantum nature of the Schrödinger equation, our hypothesis is that it may be capable of providing richer sources of information. The two main features of the Schrödinger operator that we exploit in this paper are its non-ergodicity and the presence of quantum interferences due to the existence of complex amplitudes with both positive and negative components. Our proposed graph characterization approach is based on the Fourier analysis of the quantum equivalent of the heat flow trace, thus relating frequency to structure. Our experiments, performed both on synthetic and real-world data, demonstrate that this new method can be successfully applied to the characterization of different types of graph structures.