Related papers: Some remarks on quasi-Hermitian operators
A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining inner product of the physical…
In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability conserving). To express the Hermiticity of a…
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…
In this paper we continue our analysis of intertwining relations for both self-adjoint and not self-adjoint operators. In particular, in this last situation, we discuss the connection with pseudo-hermitian quantum mechanics and the role of…
The aim of this article is to define and compare several distances (or metrics) between operators acting on different (separable) Hilbert spaces. We consider here three main cases of how to measure the distance between two bounded…
During the recent developments of quantum theory it has been clarified that the observable quantities (like energy or position) may be represented by operators (with real spectra) which are manifestly non-Hermitian. The mathematical…
We give necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss structure of such operators in the case of self-adjoint and normal operators separately.…
Within the context of non-Hermitian quantum mechanics, we use the generators of eigenvectors of the Hamiltonian to construct a unitary inner product space. Such models have been of interest in recent years, for instance, in the context of…
An absolute continuity approach to quasinormality which relates the operator in question to the spectral measure of its modulus is developed. Algebraic characterizations of some classes of operators that emerged in this context are…
In \cite{b-i-t}, F. Bagarello, A. Inoue and C. Trapani investigated some operators defined by Riesz bases. These operators connect with ${\it quasi}$-${\it hermitian \; quantum \; mechanics}$, and its relatives. In this paper, we change the…
The Moyal product is used to cast the equation for the metric of a non-hermitian Hamiltonian in the form of a differential equation. For Hamiltonians of the form $p^2+V(ix)$ with $V$ polynomial this is an exact equation. Solving this…
Standard quantum mechanics predicts the non-conservation of state norms and probability when the fundamental requirement of the Hermiticity of the Hamiltonian is relaxed. Biorthogonal quantum mechanics, or the more general metric formalism,…
Non-hermitian quantum graphs possessing real (i.e., in principle, observable) spectra are studied via their discretization. The discretized Hamiltonians are assigned, constructively, an elementary pseudometric and/or a more complicated…
We determine what should correspond to the Dirac operator on certain quantized hermitian symmetric spaces and what its properties are. A new insight into the quantized wave operator is obtained.
In this paper we introduce and prove some properties of $(\alpha;\beta)$-normal operators according to semi-Hilbertian space structures. Furthermore we s,ate various inequalities between the A-operator norm and A-numerical radius of…
PT-symmetric quantum mechanics is an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose and complex conjugate) is replaced by the physically transparent condition of space-time reflection…
Motivated by potential theory on discrete spaces, we study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. These operators are discrete analogues of the classical…
We construct a discrete non-hermitean momentum operator, which implements faithfully the non self-adjoint nature of momentum for a particle in a box. Its eigenfunctions are strictly limited to the interior of the box in the continuum limit,…
If $T$ is a semibounded self-adjoint operator in a Hilbert space $(H, \, (\cdot , \cdot))$ then the closure of the sesquilinear form $(T \cdot , \cdot)$ is a unique Hilbert space completion. In the non-semibounded case a closure is a…
This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of…