10,531 research outputs found
Pseudospectra in non-Hermitian quantum mechanics
We propose giving the mathematical concept of the pseudospectrum a central
role in quantum mechanics with non-Hermitian operators. We relate
pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint
operators, and basis properties of eigenfunctions. The abstract results are
illustrated by unexpected wild properties of operators familiar from
PT-symmetric quantum mechanics.Comment: version accepted for publication in J. Math. Phys.: criterion
excluding basis property (Proposition 6) added, unbounded time-evolution
discussed, new reference
Universal spectra of random Lindblad operators
To understand typical dynamics of an open quantum system in continuous time,
we introduce an ensemble of random Lindblad operators, which generate Markovian
completely positive evolution in the space of density matrices. Spectral
properties of these operators, including the shape of the spectrum in the
complex plane, are evaluated by using methods of free probabilities and
explained with non-Hermitian random matrix models. We also demonstrate
universality of the spectral features. The notion of ensemble of random
generators of Markovian qauntum evolution constitutes a step towards
categorization of dissipative quantum chaos.Comment: 6 pages, 4 figures + supplemental materia
Bosonization method for second super quantization
A bosonic-fermionic correspondence allows an analytic definition of
functional super derivative, in particular, and a bosonic functional calculus,
in general, on Bargmann- Gelfand triples for the second super quantization. A
Feynman integral for the super transformation matrix elements in terms of
bosonic anti-normal Berezin symbols is rigorously constructed.Comment: In memoriam of F. A. Berezin, accepted in Journal of Nonlinear
Mathematical Physics, 15 page
Simultaneous diagonalisation of the covariance and complementary covariance matrices in quaternion widely linear signal processing
Recent developments in quaternion-valued widely linear processing have
established that the exploitation of complete second-order statistics requires
consideration of both the standard covariance and the three complementary
covariance matrices. Although such matrices have a tremendous amount of
structure and their decomposition is a powerful tool in a variety of
applications, the non-commutative nature of the quaternion product has been
prohibitive to the development of quaternion uncorrelating transforms. To this
end, we introduce novel techniques for a simultaneous decomposition of the
covariance and complementary covariance matrices in the quaternion domain,
whereby the quaternion version of the Takagi factorisation is explored to
diagonalise symmetric quaternion-valued matrices. This gives new insights into
the quaternion uncorrelating transform (QUT) and forms a basis for the proposed
quaternion approximate uncorrelating transform (QAUT) which simultaneously
diagonalises all four covariance matrices associated with improper quaternion
signals. The effectiveness of the proposed uncorrelating transforms is
validated by simulations on both synthetic and real-world quaternion-valued
signals.Comment: 41 pages, single column, 10 figure
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