Computation of the spectra of some integral operators and application to the numerical solution of some linear integral equations

In this paper, we present two methods of computing the spectrum of a compact integral operator. The first method is based on an exact matrix representation of the operator. The second method uses a convenient quadrature method to discretisize the integral operator and to provide accurate approximations to the spectrum and the eigenfunctions of this later. Also, we show how our methods can be used in the framework of some stable procedures for the approximation of f^† the normal solution of the minimal L²-norm of the integral equation of the first kind Af = g, which is often an ill-posed equation. These procedures are based on a spectral expansion of the operator A. To finish, we give some numerical examples that illustrate the results of this work.     

Commutativity up to a factor for bounded and unbounded operators

In this paper, we further investigate the problem of commutativity up to a factor (or λ-commutativity) in the setting of bounded and unbounded linear operators in a complex Hilbert space. The results are based on a new approach to the problem. We finish the paper by a conjecture on the commutativity of self-adjoint operators.     

Entanglement Dynamics Induced by a Squeezed Coherent Cavity Coupled Nonlinearly with a Qubit and Filled with a Kerr-Like Medium

An analytical solution for a master equation describing the dynamics of a qubit interacting with a nonlinear Kerr-like cavity through intensity-dependent coupling is established. A superposition of squeezed coherent states is propped as the initial cavity field. The dynamics of the entangled qubit-cavity states are explored by negativity for different deformed function of the intensity-dependent coupling. We have examined the effects of the Kerr-like nonlinearity and the qubit-cavity detuning as well as the phase cavity damping on the generated entanglement. The intensity-dependent coupling increases the sensitivity of the generated entanglement to the phase-damping. The stability and the strength of the entanglement are controlled by the Kerr-like nonlinearity, the qubit-cavity detuning, and the initial cavity non-classicality. These physical parameters enhance the robustness of the qubit-cavity entanglement against the cavity phase-damping. The high initial cavity non-classicality enhances the robustness of the qubit-cavity entanglement against the phase-damping effect.     

Unbounded operators having self-adjoint,subnormal, or hyponormal powers

We show that if a densely defined closable operator A is such that the resolvent set of A² is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial $p(A)$. We also generalize a recent result by Sebestyén–Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given. One of them is a proof that if T is a quasinormal (unbounded) operator such that $T^n$ is normal for some $n\ge 2$, then T is normal. Hence a closed subnormal operator T such that $T^n$ is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that $A^p$ and $A^q$ are self-adjoint for some coprime numbers p and q, then A must be self-adjoint.