We show that a quantum system possessing an exact antilinear symmetry, in particular PT -symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT -symmetry to a Hermitian Hamiltonian. We apply our general results to PT symmetry in finite-dimensions and give the explicit form of the...

For a given linear matrix function A1−B1XB 1 , where X is a variable Hermitian matrix, this paper derives a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the matrix function subject to a pair of consistent matrix equations B2XB ∗ 2 = A2 and B3XB ∗ 3 = A3. As applications, we give necessary and sufficient conditions for the triple matrix equat...

In the context of quasi-Hermitian theories, which are non-Hermitian in the conventional sense, but can be made Hermitian by the introduction of a dynamically-determined metric η, we address the problem of how the functional integral and the Feynman diagrams deduced therefrom “know” about the metric. Our investigation is triggered by a result of Bender, Chen and Milton, who calculated perturbati...

We consider a complex Hermitian manifold of complex dimensions four with a Hermitian metric and a Chern connection. It is shown that the action that determines the dynamics of the metric is unique, provided that the linearized Einstein action coupled to an antisymmetric tensor is obtained, in the limit when the imaginary coordinates vanish. The unique action is of the Chern-Simons type when exp...

In quantum mechanics, predictions are made by way of calculating expectation values of observables, which take the form of Hermitian operators. Non-Hermitian operators, however, are not necessarily devoid of physical significance, and they can play a crucial role in the characterization of quantum states. Here we show that the expectation values of a particular set of non-Hermitian matrices, wh...

A structure preserving Sort-Jacobi algorithm for computing eigenvalues or singular values is presented. The proposed method applies to an arbitrary semisimple Lie algebra on its (−1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra an...

A plane is a two-dimensional right vector space V over a division algebra D, which we assume in this paper has an involution; a hermitian form h : V × V −→ D is hyperbolic, if, with respect to a properly choosen basis, it is given by the matrix 0 1 1 0 ) ; the pair (V, h) is called a hyperbolic plane. If D is central simple over a number field K, a hyperbolic plane (V, h) gives rise to a reduct...

Abstract. Recently developed methods for PT-symmetric models can be applied to quantum-mechanical matrix and vector models. In matrix models, the calculation of all singlet wave functions can be reduced to the solution a one-dimensional PT-symmetric model. The large-N limit of a wide class of matrix models exists, and properties of the lowest-lying singlet state can be computed using WKB. For m...

in this paper, we are going to study the g-natural metrics on the tangent bundle of finslermanifolds. we concentrate on the complex and kählerian and hermitian structures associated with finslermanifolds via g-natural metrics. we prove that the almost complex structure induced by this metric is acomplex structure on tangent bundle if and only if the finsler metric is of scalar flag curvature. t...