MHD α2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator

Günther, Uwe; Stefani, Frank; Znojil, Miloslav

doi:10.1063/1.1915293

Cited by 71 publications

(102 citation statements)

References 84 publications

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“…In all of these cases a non-Hermitian Hamiltonian having a real spectrum appeared mysterious at the time, but now the explanation is clear: In every case the nonHermitian Hamiltonian is P T symmetric. Hamiltonians having P T symmetry have also been used to describe magnetohydrodynamic systems [34,37] and to study nondissipative time-dependent systems interacting with electromagnetic fields [30].…”

Section: P T Quantum Mechanicsmentioning

confidence: 99%

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Bender¹,

Brody²

2009

Lecture Notes in Physics

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The shortest path between two truths in the real domain passes through the complex domain. -Jacques Hadamard, The Mathematical Intelligencer 13 (1991) Consider the set of all Hamiltonians whose largest and smallest energy eigenvalues, E max and E min , differ by a fixed energy ω. Given two quantum states, an initial state |ψ I and a final state |ψ F , there exist many Hamiltonians H belonging to this set under which |ψ I evolves in time into |ψ F . Which Hamiltonian transforms the initial state to the final state in the least possible time τ ? For Hermitian Hamiltonians, τ has a nonzero lower bound. However, among complex non-Hermitian P T -symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of τ can be made arbitrarily small because for P T -symmetric Hamiltonians the evolution path from the vector |ψ I to the vector |ψ F , as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here resembles the effect in general relativity in which two space-time points can be made arbitrarily close if they are connected by a wormhole. This result may have applications in quantum computing.

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Section: P T Quantum Mechanicsmentioning

confidence: 99%

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Bender¹,

Brody²

2009

Lecture Notes in Physics

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“…When ≥ 0, the PT symmetry is unbroken, but when < 0, the PT symmetry is broken ( figure 1). Moreover, some extensions of non-Hermitian PT -symmetric Hamiltonians H given by equation (2.1) have also been considered [15][16][17][18][19][20]. The notion of pseudo-Hermiticity has also been introduced [21][22][23][24] and it has been shown that every Hamiltonian with a real spectrum is pseudo-Hermitian [25].…”

Section: Non-hermitian Pt -Symmetric Hamiltoniansmentioning

confidence: 99%

Complex PT -symmetric nonlinear Schrödinger equation and Burgers equation

Yan

2013

Phil. Trans. R. Soc. A.

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One contribution of 17 to a Theme Issue 'PT quantum mechanics' . The complex PT -symmetric nonlinear wave models have drawn much attention in recent years since the complex PT -symmetric extensions of the Kortewegde Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex PT -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex PT symmetry. We then report on exact solutions of one-and twodimensional nonlinear Schrödinger equations (known as the Gross-Pitaevskii equation in Bose-Einstein condensates) with several complex PT -symmetric potentials. Finally, some complex PT -symmetric extension principles are used to generate some complex PT -symmetric nonlinear wave equations starting from both PT -symmetric (e.g. the KdV equation) and non-PT -symmetric (e.g. the Burgers equation) nonlinear wave equations. In particular, we discuss exact solutions of some representative ones of the complex PT -symmetric Burgers equation in detail.

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“…The matrix valued function M is called the Weyl function of A corresponding to the boundary triple 6) and hence M (z) is well defined and takes values in C d×d . It follows from the identity that the Weyl function M (λ) satisfies the identities:…”

Section: Definition 42mentioning

confidence: 99%

“…K induces in an obvious way a sign type spectrum for linear operators. In the last two decades, this notion was frequently used in theoretical physics in connection with PT -symmetric problems; here, we mention only [3][4][5][6][7] and in the study of PT -symmetric operators, we refer to [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Coupling of definitizable operators in Krein spaces

Derkach¹,

Trunk²

2017

Nanosystems: Phys. Chem. Math.

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Indefinite Sturm-Liouville operators defined on R are often considered as a coupling of two semibounded symmetric operators defined on R + and R − , respectively. In many situations, those two semibounded symmetric operators have in a special sense good properties like a Hilbert space self-adjoint extension.In this paper, we present an abstract approach to the coupling of two (definitizable) self-adjoint operators. We obtain a characterization for the definitizability and the regularity of the critical points. Finally we study a typical class of indefinite Sturm-Liouville problems on R.

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MHD α2-dynamo, Squire equation and PT-symmetric interpolation between square well and harmonic oscillator

Cited by 71 publications

References 84 publications

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Complex PT -symmetric nonlinear Schrödinger equation and Burgers equation

Coupling of definitizable operators in Krein spaces

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