Mathematical and physical aspects of complex symmetric operators

Garcia, Stephan Ramon; Prodan, Emil; Putinar, Mihai

doi:10.1088/1751-8113/47/35/353001

Cited by 138 publications

(89 citation statements)

References 157 publications

(270 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We do not intend to cover all available results of this extremely vast field, but rather to systematize the material relevant for description of nonlinear systems presented in the subsequent sections. For comprehensive reviews on non-Hermitian operators in physics and mathematics, in addition to the works listed in the Introduction we also mention the reviews by Cannata, Dedonder, and Ventura (2007); Daley (2014); Garcia, Prodan, and Putinar (2014); Muga et al (2004);and Rotter (2009), as well as the monograph of Moiseyev (2011).…”

Section: Non-hermitian Operators With Real Spectramentioning

confidence: 99%

Nonlinear waves inPT-symmetric systems

2016

View full text Add to dashboard Cite

Recent progress on nonlinear properties of parity-time (PT -) symmetric systems is comprehensively reviewed in this article. PT symmetry started out in non-Hermitian quantum mechanics, where complex potentials obeying PT symmetry could exhibit all-real spectra. This concept later spread out to optics, Bose-Einstein condensates, electronic circuits, and many other physical fields, where a judicious balancing of gain and loss constitutes a PT -symmetric system. The natural inclusion of nonlinearity into these PT systems then gave rise to a wide array of new phenomena which have no counterparts in traditional dissipative systems. Examples include the existence of continuous families of nonlinear modes and integrals of motion, stabilization of nonlinear modes above PTsymmetry phase transition, symmetry breaking of nonlinear modes, distinctive soliton dynamics, and many others. In this article, nonlinear PT -symmetric systems arising from various physical disciplines are presented; nonlinear properties of these systems are thoroughly elucidated; and relevant experimental results are described. In addition, emerging applications of PT symmetry are pointed out.

show abstract

Section: Non-hermitian Operators With Real Spectramentioning

confidence: 99%

Nonlinear waves inPT-symmetric systems

2016

View full text Add to dashboard Cite

show abstract

“…Since C is a conjugation on H, by [18,Lem. 2.11], there exists an orthonormal basis {e n } such that Ce n = e n for all n. For each n ≥ 1, denote by P n the projection of H onto ∨{e i : 1 ≤ i ≤ n}.…”

Section: 1mentioning

confidence: 99%

The orthogonal Lie algebra of operators: Ideals and derivations

Zhu

2020

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

We study in this paper the infinite-dimensional orthogonal Lie algebra O C which consists of all bounded linear operators T on a separable, infinite-dimensional, complex Hilbert space H satisfying CT C = −T * , where C is a conjugation on H. By employing results from the theory of complex symmetric operators and skew-symmetric operators, we determine the Lie ideals of O C and their dual spaces. We study derivations of O C and determine their spectra. These results complete some results of P. de la Harpe and provide interesting contrasts between O C and the algebra B(H) of all bounded linear operators on H. IntroductionThe study of skew-symmetric matrices (i.e., those matrices M satisfying M + M tr = 0, where M tr denotes the transpose of M ) can be traced back to the work of L.-K. Hua on automorphic functions [27,28], N. Jacobson on projective geometry [30], and C. Siegel on symplectic geometry [48]. Skew-symmetric matrices arise naturally in partial differential equations [42], differential geometry [13], algebraic geometry [15] and many other mathematical disciplines. In the finite-dimensional Hilbert space of even dimension, there is a natural one to one correspondence between the set of all non-singular skew-symmetric matrices and the set of all sympletic forms on the Hilbert space ([32, page 16]). As a classical finite-dimensional Lie algebra over the complex field C, the orthogonal Lie algebra so(n, C) consists of all n × n skew-symmetric complex matrices, that is, so(n, C) = {X ∈ M n (C) : X + X tr = 0}, where M n (C) denotes the set of all n × n complex matrices. It is known that so(n, C) is the Lie algebra of the complex orthogonal Lie group SO(n, C) = {X ∈ M n (C) : XX tr = I, detX = 1} (see [23, page 41] or [25, page 341] for more details), and plays important roles in the classification of semi-simple Lie algebras.Skew-symmetric matrices are playing important roles in quantum physics. In particular, they are closely related to the noncommutative spacetimes. A noncommutative spacetime [49] is defined by the Hermitian generatorsx i of a noncommutative C * -algebra of "functions on spacetime" which obey the commutation

show abstract

“…Unbounded examples appear in the complex scaling theory for Schrödinger operators, certain non-self-adjoint boundary value problems, and -symmetric quantum theory [1].…”

Section: Stephan Ramon Garcia Communicated By Steven J Miller and Cementioning

confidence: 99%