ON THE REAL MATRIX REPRESENTATION OF PT-SYMMETRIC OPERATORS

Non-Hermitian matrix description of the PT −symmetric anharmonic oscillators

The PT −symmetric differential Schrödinger equation Hψ = Eψ with the operator H = H(x) = p + a x + i β x + c x + i δ x ≡ H∗(−x) on L2(−∞,∞) is studied. At a > 0 it is re-arranged as a linear algebraic diagonalization. With rigorous proof, our non-variational construction of bound states offers an infinite-dimensional analogue to the recent finite-dimensional quasi-exact solution available at th...

On the solving matrix equations by using the spectral representation

‎The purpose of this paper is to solve two types of Lyapunov equations and quadratic matrix equations by using the spectral representation‎. ‎We focus on solving Lyapunov equations $AX+XA^*=C$ and $AX+XA^{T}=-bb^{T}$ for $A‎, ‎X in mathbb{C}^{n times n}$ and $b in mathbb{C} ^{n times s}$ with $s < n$‎, ‎which $X$ is unknown matrix‎. ‎Also‎, ‎we suggest the new method for solving quadratic matri...

Matrix Representation of Operators Using Frames

In this paper it is investigated how to find a matrix representation of operators on a Hilbert space H with Bessel sequences, frames and Riesz bases. In many applications these sequences are often preferable to orthonormal bases (ONBs). Therefore it is useful to extend the known method of matrix representation by using these sequences instead of ONBs for these application areas. We will give ba...

Effect of Relative Phase on the Stability of Temporal Bright Solitons in a PT- Symmetric NLDC

In this paper we numerically investigate the effect of relative phase on thestability of temporal bright solitons in a Nano PT- Symmetric nonlinear directionalcoupler (NLDC) by considering gain in bar and loss in cross. We also study the effect ofrelative phase on the output perturbed bright solitons energies, in the range of   0 to 180 . By using perturbation theory three eigenfunctions an...

Frames and Matrix Representation of Operators