Complex Extension of Quantum Mechanics

Bender, Carl M.; Brody, Dorje C.; Jones, H. F.

doi:10.1103/physrevlett.89.270401

Cited by 1,712 publications

(1,264 citation statements)

References 22 publications

Supporting

Mentioning

1,244

Contrasting

Unclassified

Order By: Relevance

“…Examples of this kind, which were meanwhile also realized in the experiment [6][7][8][9][10] , are the unidirectional invisibility of a gain-loss potential 11 , devices that can simultaneously act as laser and as a perfect absorber [12][13][14] and resonant structures with unusual features like non-reciprocal light transmission 10 or loss-induced lasing [15][16][17] . In particular, systems with a so-called parity-time (PT ) symmetry 18 , where gain and loss are carefully balanced, have recently attracted enormous interest in the context of nonHermitian photonics [19][20][21][22][23][24] .…”

mentioning

confidence: 99%

Constant-intensity waves and their modulation instability in non-Hermitian potentials

et al. 2015

View full text Add to dashboard Cite

In all of the diverse areas of science where waves play an important role, one of the most fundamental solutions of the corresponding wave equation is a stationary wave with constant intensity. The most familiar example is that of a plane wave propagating in free space. In the presence of any Hermitian potential, a wave's constant intensity is, however, immediately destroyed due to scattering. Here we show that this fundamental restriction is conveniently lifted when working with non-Hermitian potentials. In particular, we present a whole class of waves that have constant intensity in the presence of linear as well as of nonlinear inhomogeneous media with gain and loss. These solutions allow us to study the fundamental phenomenon of modulation instability in an inhomogeneous environment. Our results pose a new challenge for the experiments on non-Hermitian scattering that have recently been put forward.

show abstract

mentioning

confidence: 99%

Constant-intensity waves and their modulation instability in non-Hermitian potentials

et al. 2015

View full text Add to dashboard Cite

show abstract

“…This follows from the fact that they apparently coincide on solutions of the boundary value problem (1-2) which form a basis in L 2 (−π, π) and both are sesquilinear. We infer, hence, that any such a Hamiltonian has positive CPT -normalizable eigenfunctions; B) CPT completeness condition (see [2]) is equivalent to the usual completeness condition for a non-selfadjoint Hamiltonian in the space L 2 (−π, π) given by (21); C) CPT extension of quantum mechanics for such Hamiltonians should follow the same lines already reported in [2] but as long as the situation with the continuous spectrum is unclear this extension is incomplete.…”

Section: IImentioning

confidence: 88%

“…This has given rise to the possibility of constructing a complex extension of quantum mechanics [2]. Before the discovery of the C operator [2] the main difficulty in constructing a self-consistent complex extension of quantum mechanics was the presence of negative PT -norms for some PT -symmetric Hamiltonians. Using the CPT operation a new norm was defined [2] and it was shown to be positive for some models.…”

mentioning

confidence: 99%

Is the norm always positive?

Samsonov

Roy

2005

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

Abstract. We give an explicit example of an exactly solvable PT -symmetric Hamiltonian with the unbroken PT symmetry which has one eigenfunction with the zero PT -norm. The set of its eigenfunctions is not complete in corresponding Hilbert space and it is non-diagonalizable. In the case of a regular Sturm-Liouville problem any diagonalizable PT -symmetric Hamiltonian with the unbroken PT symmetry has a complete set of positive CPT -normalazable eigenfunctions. For non-diagonalizable Hamiltonians a complete set of CPT -normalazable functions is possible but the functions belonging to the root subspace corresponding to multiple zeros of the characteristic determinant are not eigenfunctions of the Hamiltonian anymore.1. In recent years it has been shown that non Hermitian Hamiltonians, in particular the PT -symmetric ones may have real eigenvalues [1]. This has given rise to the possibility of constructing a complex extension of quantum mechanics [2]. Before the discovery of the C operator [2] the main difficulty in constructing a self-consistent complex extension of quantum mechanics was the presence of negative PT -norms for some PT -symmetric Hamiltonians. Using the CPT operation a new norm was defined [2] and it was shown to be positive for some models. In this Letter we would like to stress that this is true only if the PT -norm is non-zero. Here we shall consider an explicit example of an exactly solvable PT -symmetric Hamiltonian one of whose eigenfunction has the zero PT -norm thus proving that such a situation may occur in a PT -symmetric Hamiltonian. Further, using the fact that the PT operation applied to an eigenfunction of the given Hamiltonian may only change its sign resulting in the property that the CPT operation inverts the sign of a PT norm if it is negative, we conclude that application of the CPT operation to an eigenfunction of a PT symmetric Hamiltonian is equivalent to first going to an eigenfunction of the adjoint operator and then taking the complex conjugation provided the given solution is normalized properly. Subsequently we show that for a regular Sturm-Liouville problem the zero CPT -norms may appear only if the characteristic determinant has multiple zeros. In this case the system of eigenfunctions is not complete in corresponding Hilbert space implying that the Hamiltonian related with such a problem is non-diagonalizable. We show that both for diagonalizble and non-diagonalizable Hamiltonians one is able to define a Hilbert space with a positive CPT -norm but in the later case the basis functions corresponding to a degenerate root subspace are not eigenfunctions of the Hamiltonian.

show abstract

“…At β = β c , the real eigenvalue responsible for the kink's instability vanishes. Actually, β c satisfies inequality Equation (15), i.e., the existence range is smaller than the range of the background stability whenever Equations (16) and (17) hold. Figure 3 shows the dependence of the real part of the instability eigenvalues on β for fixed = 0.25.…”

Section: Instability Of the Kk And Ka Complexes At >mentioning

confidence: 99%

A PT -Symmetric Dual-Core System with the Sine-Gordon Nonlinearity and Derivative Coupling

2016

View full text Add to dashboard Cite

Abstract:As an extension of the class of nonlinear P T -symmetric models, we propose a system of sine-Gordon equations, with the P T symmetry represented by balanced gain and loss in them. The equations are coupled by sine-field terms and first-order derivatives. The sinusoidal coupling stems from local interaction between adjacent particles in coupled Frenkel-Kontorova (FK) chains, while the cross-derivative coupling, which was not considered before, is induced by three-particle interactions, provided that the particles in the parallel FK chains move in different directions. Nonlinear modes are then studied in this system. In particular, kink-kink (KK) and kink-anti-kink (KA) complexes are explored by means of analytical and numerical methods. It is predicted analytically and confirmed numerically that the complexes are unstable for one sign of the sinusoidal coupling and stable for another. Stability regions are delineated in the underlying parameter space. Unstable complexes split into free kinks and anti-kinks that may propagate or become quiescent, depending on whether they are subject to gain or loss, respectively.

show abstract

Complex Extension of Quantum Mechanics

Cited by 1,712 publications

References 22 publications

Constant-intensity waves and their modulation instability in non-Hermitian potentials

Constant-intensity waves and their modulation instability in non-Hermitian potentials

Is the norm always positive?

A PT -Symmetric Dual-Core System with the Sine-Gordon Nonlinearity and Derivative Coupling

Contact Info

Product

Resources

About