We establish that a perfect-transmission scattering problem can be described by a class of parity and time reversal symmetric operators and hereby we provide a scenario for understanding and implementing the corresponding quasi-Hermitian quantum mechanical framework from the physical viewpoint. One of the most interesting features of the analysis is that the complex eigenvalues of the underlying non-Hermitian problem, associated with a reflectionless scattering system, lead to the loss of perfect-transmission energies as the parameters characterizing the scattering potential are varied. On the other hand, the scattering data can serve to describe the spectrum of a large class of Schrödinger operators with complex Robin boundary conditions.

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We develop a framework to test the Equivalence Principle (EP) under conditions where the quantum aspects of nature cannot be neglected, specifically in the context of interference phenomena with unstable particles. We derive the nonrelativistic quantum equation that describes the evolution of the wavefunction of unstable particles under the assumption of the validity of the EP and when small deviations are assumed to occur. As an example, we study the propagation of unstable particles in a COW experiment, and we briefly discuss the experimental implications of our formalism. MotivationThe Equivalence Principle (EP) plays a central role in our understanding of nature. It lies at the basis not only of the general theory of relativity, but of the notion of inertia itself, since the practical realization of an inertial frame is only possible by assuming the EP. It is thus not surprising that testing such a principle should have a tradition that is almost as old as modern physics. Experimental programs have sought to explore its validity for the most diverse kinds of physical systems and conditions [1][2][3][4][5][6][7][8] ranging from space-bound experiments to tests using antimatter [9][10][11][12]. One of the aspects where the principle has been least explored, perhaps due to technical difficulties, is the quantum realm, that is, those situations where the quantum/gravity interface becomes an essential aspect of the experiment. Among the most conspicuously quantum phenomena are those associated with unstable particles, and hence the focus of this paper will be tests of the EP with unstable quantum systems.Among the limited instances where the quantum/gravity interface has been explored are the well-known COW experiments [13] and the recent cold neutron experiments [14]. Their results, which are in accordance with our theoretical expectations, have led to a degree of confusion resulting from an imprecise statement of the principle one wishes to test [15][16][17]. In the classical context one of the simplest versions of the EP is the universality of free fall. This is often stated as the independence of the acceleration of test objects on their initial velocity, mass or composition [18][19][20][21][22][23][24][25][26][27][28], i.e., the object's acceleration should depend only on its location. In the quantum context it is clear that the terms must be modified, not only because the objects cannot have precise position and initial velocity, but because, even in the absence of gravity, the mass of a particle also determines how the wavepacket spreads (see also Refs. [29,30]).A recent discussion of such issues can be found in Ref.[31] where four broad categories of the EP are considered, two of which are suitable for applicability to the quantum context. The difference between the two is that one refers to a situation where the gravitational field can be considered homogeneous (isotropic and time independent), and the other involves deviations from that. The version we use is the first which can be stated as fol...

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