In this paper we construct infinitely many selfadjoint solutions of the control algebraic Riccati equation using invariant subspaces of the associated Hamiltonian. We do this under the assumption that the system operator is normal and has compact inverse, and that the Hamiltonian possesses a Riesz basis of invariant subspaces.

We give a simple proof of the fact that every diagonalizable operator that has a real spectrum is quasi-Hermitian and show how the metric operators associated with a quasiHermitian Hamiltonian are related to the symmetry generators of an equivalent Hermitian Hamiltonian. PACS number: 03.65.-w

The proof of the Khalfin Theorem for neutral meson complex is analyzed. It is shown that the unitarity of the time evolution operator for the total system under considerations assures that the Khalfin's Theorem holds. The consequences of this Theorem for the neutral mesons system are discussed: it is shown, eg., that diagonal matrix elements of the exact effective Hamiltonian for the neutral me...

We show that elliptic Calogero-Moser system and its Lax operator found by Krichever can be obtained by hamiltonian reduction from the integrable hamiltonian system on the cotangent bundle to the central extension of the algebra of slN (C) currents. ∗email: [email protected] [email protected]

A novel expansion of the evolution operator associated with a – in general, time-dependent – perturbed quantum Hamiltonian is presented. It is shown that it has a wide range of possible realizations that can be fitted according to computational convenience or to satisfy specific requirements. As a remarkable example, the quantum Hamiltonian describing a laser-driven trapped ion is studied in de...

We analyze the proof of the Khalfin Theorem for neutral meson complex. The consequences of this Theorem are discussed: using this Theorem we find, eg., that diagonal matrix elements of the exact effective Hamiltonian for the neutral meson complex can not be equal if CPT symmetry holds and CP symmetry is violated. The Properties of time evolution governed by a time–independent effective Hamilto-...

We decompose the Hilbert space of wave functions into two subspaces, and assign to a given observable two effective representatives that act in the model space. The first serves to determine some of the eigen-values of the full observable, while the second serves to determine its matrix elements, in any basis in one of the subspaces, in terms of quantities pertaining to the model space. We also...

Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single Master constraint. The MCP is designed to overcome the complications associated with the non – Lie – algebra structure of the Dirac algebra of Hamiltonian constraints and was successfully tested in various field theory models. For t...

One of the simplest non-Hermitian Hamiltonians, first proposed by Schwartz in 1960, that may possess a spectral singularity is analysed from the point of view of the non-Hermitian generalization of quantum mechanics. It is shown that the η operator, being a second-order differential operator, has supersymmetric structure. Asymptotic behaviour of the eigenfunctions of a Hermitian Hamiltonian equ...

It is argued that renormalisation group flow can be interpreted as being a Hamiltonian vector flow on a phase space which consists of the couplings of the theory and their conjugate “momenta”, which are the vacuum expectation values of the corresponding composite operators. The Hamiltonian is linear in the conjugate variables and can be identified with the vacuum expectation value of the trace ...