During the recent developments of quantum theory it has been clarified that the observable quantities (like energy or position) may be represented by operators Λ (with real spectra) which are manifestly non-Hermitian in a preselected "friendly" Hilbert space H (F ) . The consistency of these models is known to require an upgrade of the inner product, i.e., mathematically speaking, a transition H (F ) → H (S) to another, "standard" Hilbert space.We prove that whenever we are given more than one candidate for an observable (i.e., say, two operators Λ 0 and Λ 1 ) in advance, such an upgrade need not exist in general.
The realization of a genuine phase transition in quantum mechanics requires
that at least one of the Kato's exceptional-point parameters becomes real. A
new family of finite-dimensional and time-parametrized quantum-lattice models
with such a property is proposed and studied. All of them exhibit, at a real
exceptional-point time $t=0$, the Jordan-block spectral degeneracy structure of
some of their observables sampled by Hamiltonian $H(t)$ and site-position
$Q(t)$. The passes through the critical instant $t=0$ are interpreted as
schematic simulations of non-equivalent versions of the Big-Bang-like quantum
catastrophes.Comment: 17 pp., 7 figure
Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode selective losses, and minimal quantum systems, and the meteoric research on them has mainly focused on the wide range of novel functionalities they demonstrate. Here, we address the following questions: Does anything remain constant in the dynamics of such open systems? What are the consequences of such conserved quantities? Through spectral-decomposition method and explicit, recursive procedure, we obtain all conserved observables for general
PT
-symmetric systems. We then generalize the analysis to Hamiltonians with other antilinear symmetries, and discuss the consequences of conservation laws for open systems. We illustrate our findings with several physically motivated examples.
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