KeyNote Speaker Committee
  • Prof. Mauricio Porto Pato
  • Prof. Mauricio Porto Pato
  • Universidade de São Paulo, Brazil
  • Title: Pseudo-Hermitian random matrices
  • 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.
  • Biography: Dr Pato was born in 1944 at the city of Salvador capital of the Brazilian northeast state of Bahia. He finished his undergraduate studies at the Federal University of Bahia and obtained his Master degree at the University of Brasilia at the capital of Brazil. He then moved to the state of S?o Paulo where he got his PhD at the University of Sao Paulo (USP), studying statistical properties of the electons inside a vacuum pump of the Orbitron type used in the Pelletron Accelerator of the nuclear department. After a short pos-doc stage at the the University of Wisconsin, US, he got a permanent position at the University of S?o Paulo where he has been working since then. After a period in which he was more dedicated to teaching, he started a research career working in different areas of physics. In nuclear physics he developed a semi-classical formalism to separate the diffractive and the refractive components of the elastic scattering amplitude in a heavy ion collision. With collaborators, he pointed out the importance of break-up in the fusion cross-section of halo nuclei. In the field of accelerators, he developed with M. S. Hussein of USP, a principle of acceleration of charged particles that uses the intense field of a laser coupled to a tranverse static field whose sign are appropriately switched. In the early 90's he began the study of random matrix theories which then became his major interest. He then collaborated with important groups of this field like the TUNL group of G. Mitchell, in US, and, in France, the LPTMS group of the Paris-Sud University lead by O. Bohigas. In the University of S?o Paulo, he constructed with M. S. Hussein an ensemble of random matrices denoted by them as deformed Gaussian ensemble with the objective of describing partial symmetry conservation. This ensemble was successfully applied to analise experimental data on the conservation of the isospin symmetry measured by the TUNL group. Several applications of the deformed ensemble then followed as for instance to the statistics observed at the critical point of the Anderson transition. In the last decade, he collaborated mostly with the LPTMS group. As a result of this collaboration, A formalism to deal with missing levels was obtained which gave, as a side product, a model that interpolates between regularity and chaos. Also, the concept of disorder was introduced in random matrix theories and applied to growth process in a random media. Finally, more recently, the interest moved to the study of non-Hermitian random operators, in particular, to the investigation of the class of pseudo-Hermitian matrices which are associated to PT symmetric systems, that is, systems invariant under the combined parity (P) and time-reversal (T) transformations.
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April 14-16, 2015
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December 11, 2014  >> February 16, 2015
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