Linking Scalar Elastodynamics and Non-Hermitian Quantum Mechanics

Recent years have seen a fascinating pollination of ideas from quantum theories to elastodynamics---a theory that phenomenologically describes the time-dependent macroscopic response of materials. Here, we open a route to transfer additional tools from non-Hermitian quantum mechanics. We begin by identifying the differences and similarities between the one-dimensional elastodynamics equation without body forces and the time-independent Schr\"odinger equation and finding the condition under which the two are equivalent. Subsequently, we demonstrate the application of the non-Hermitian perturbation theory to determine the response of elastic systems; the calculation of leaky modes and the energy decay rate in heterogenous solids with open boundaries using a quantum mechanics approach; and construction of degeneracies in the spectrum of these assemblies. The latter result may be of technological significance, as it introduces an approach to harness extraordinary wave phenomena associated with non-Hermitian degeneracies for practical devices, by designing them in simple elastic systems. As an example of such an application, we demonstrate how an assembly of elastic slabs that is designed with two degenerate shear states according to our scheme can be used for mass sensing with enhanced sensitivity by exploiting the unique topology near the exceptional point of degeneracy.