Abstract: |
By the end of the 90’s, systems described by complex Hamiltonian
whose spectra are real attracted a great atention. It then has been
identified that this occurs when the Hamiltonian is invariant under the
combined parity (P) and time-reversal (T) transformations. This lead
to an extension of quantum mechanics to include these PT symmetic
systems. From a mathematical point of view, it has been established
that this kind of Hamiltonian belong to a class of non-Hermitian operators,
now called pseudo-Hermitians, which are connected to their
adjoints by a similarity transformation. In other words, they share
with their adjoints the same set of eigenvalues. By the same time, that
is the turning of the century, in the contaxt of random matrix theory,
it has been discovered that by performing a sequence of Householder
transformations, the Gaussian matrices are reduced to a tridiagonal
form in which the Dyson beta index can assume any real value. This
finding defined the so-called β-ensemble. In the Gausssian form, this
index has the integral values 1,2 and 4 and defines the three classes
of matrices, namely the orthogonal (GOE), the unitary (GUE), and
the symplectic (GSE) with real, complex, and quaternion elements,
respectively. In fact, the application of which one of these three classes
is used to analise statistical properties of a given physical system depends
on the behavior of the system with respect to the time-reversal
transformation. Therefore, it seems natural to investigate if an ensemble
of random matrices can be constructed to model PT symmetric
systems. In my talk, I intend to show that the special properties tridiagonal
matrices have, makes the β-ensemble a natural candidate to
discuss aspects of pseudo-Hermitian operators. I will discuss results
already published but, also, new ones under investigation, obtained
introducing non-Hermiticity in the β-ensemble. |