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1.
We discuss three Hamiltonians, each with a central-field part H0 and a PT-symmetric perturbation igz. When H0 is the isotropic Harmonic oscillator the spectrum is real for all g because H is isospectral to H0+g2/2. When H0 is the Hydrogen atom then infinitely many eigenvalues are complex for all g. If the potential in H0 is linear in the radial variable r then the spectrum of H exhibits real eigenvalues for 0<g<gc and a PT phase transition at gc. 相似文献
2.
Based on the PT-symmetric quantum theory, the concepts of PT-frame, PT-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed. It is proved that the spectrum and point spectrum of a PT-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken PT-symmetric operator are real. For a linear operator H on Cd, it is proved that H has unbroken PT- symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame on K, a new positive inner product on K is induced and called CPT-inner product. Te relationship between the CPT-adjoint and the Dirac adjoint of a densely defined linear operator is derived, and it is proved that an operator which has a bounded CPT-frame is CPT-Hermitian if and only if it is T-symmetric, in that case, it is similar to a Hermitian operator. The existence of an operator C consisting of a CPT-frame is discussed. These concepts and results will serve a mathematical discussion about PT-symmetric quantum mechanics. 相似文献
3.
We discuss space-time symmetric Hamiltonian operators of the form H=H0+igH′, where H0 is Hermitian and g real. H0 is invariant under the unitary operations of a point group G while H′ is invariant under transformation by elements of a subgroup G′ of G. If G exhibits irreducible representations of dimension greater than unity, then it is possible that H has complex eigenvalues for sufficiently small nonzero values of g. In the particular case that H is parity-time symmetric then it appears to exhibit real eigenvalues for all 0<g<gc, where gc is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether H may exhibit real or complex eigenvalues for g>0. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries. 相似文献
4.
5.
《Physics letters. A》2020,384(14):126285
We study classification of anti-Hermitian topological insulators based on the discrete symmetries: time-reversal, particle-hole and chiral symmetries. Contrary to the most general form of non-Hermitian systems, bulk boundary correspondence can hold in anti-Hermitian topological systems. We map a topologically nontrivial Hermitian Hamiltonians into an anti-Hermitian system and we show that the standard table of topological insulators can be used for anti-Hermitian Hamiltonians. 相似文献
6.
We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit. 相似文献
7.
Hiroshi Miki 《Physics letters. A》2011,376(2):65-69
A set of r non-Hermitian oscillator Hamiltonians in r dimensions is shown to be simultaneously diagonalizable. Their spectra are real and the common eigenstates are expressed in terms of multiple Charlier polynomials. An algebraic interpretation of these polynomials is thus achieved and the model is used to derive some of their properties. 相似文献
8.
WEILian-Fu WANGShun-jin 《理论物理通讯》2001,35(1):15-18
A convenient method to exactly solve the quantum-nonautonomous systems with non-Hermitian Hamiltonians is proposed.It is shown that a nonadiabatic complete biorthonormal set can be easily obtained by the gauge transformation method in which the algebraic structure of systems has been used.The nonuitary evolution operator is also found by choosing a special gauge function.All auxiliry parameters introduced in the present approach are only determined by some algebraic equations.The dynamics of two quantum-nonautonomous systems ruled by non-Hermitian Hamiltonians,including a two-photon ionization process involving two-state only and a mesoscopic RLC circuit with a source,are treated as the demonstration of our general approach. 相似文献
9.
Ali Mostafazadeh 《Pramana》2009,73(2):269-277
We present an evaluation of some recent attempts to understand the role of pseudo-Hermitian and $
\mathcal{P}\mathcal{T}
$
\mathcal{P}\mathcal{T}
-symmetric Hamiltonians in modelling unitary quantum systems and elaborate on a particular physical phenomenon whose discovery
originated in the study of complex scattering potentials. 相似文献
10.
The construction of conserved quantities for linearly coupled oscillators and study of symmetries about the conserved quantities
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In this paper, the conserved quantities are constructed using two
methods. The first method is by making an ansatz of the conserved
quantity and then using the definition of Poisson bracket to obtain
the coefficients in the ansatz. The main procedure for the second
method is given as follows. Firstly, the coupled terms in Lagrangian
are eliminated by changing the coordinate scales and rotating the
coordinate axes, secondly, the conserved quantities are obtain in
new coordinate directly, and at last, the conserved quantities are
expressed in the original coordinates by using the inverse transform
of the coordinates. The Noether symmetry and Lie symmetry of the
infinitesimal transformations about the conserved quantities are
also studied in this paper. 相似文献
11.
A class of non-Hermitian d-dimensional Hamiltonians with position-dependent mass and their η-pseudo-Hermiticity generators is presented. Illustrative examples are given in 1D, 2D, and 3D for different position-dependent mass settings. 相似文献
12.
A. Leviatan 《Progress in Particle and Nuclear Physics》2011,66(1):93-143
This overview focuses on the notion of partial dynamical symmetry (PDS), for which a prescribed symmetry is obeyed by a subset of solvable eigenstates, but is not shared by the Hamiltonian. General algorithms are presented to identify interactions, of a given order, with such intermediate symmetry structure. Explicit bosonic and fermionic Hamiltonians with PDS are constructed in the framework of models based on spectrum generating algebras. PDSs of various types are shown to be relevant to nuclear spectroscopy, quantum phase transitions and systems with mixed chaotic and regular dynamics. 相似文献
13.
传统量子系统的哈密顿是自伴算子,哈密顿的自伴性不仅保证系统遵循酉演化和保持概率守恒,而且也保证了它自身具有实的能量本征值,这类系统称为自伴量子系统.然而,确实存在一些物理系统(如PT-对称量子系统),其哈密顿不是自伴的,这类系统称为非自伴量子系统.为了深入研究PT-对称量子系统,并考虑到算子PT的共轭线性性,首先讨论了共轭线性算子的一些性质,包括它们的矩阵表示和谱结构等;其次,分别研究了具有共轭线性对称性和完整共轭线性对称性的线性算子,通过它们的矩阵表示,给出了共轭线性对称性和完整共轭线性对称性的等价刻画;作为应用,得到了关于PT-对称及完整PT-对称算子的一些有趣性质,并通过一些具体例子,说明了完整PT-对称性对张量积运算不具有封闭性,同时说明了完整PT-对称性既不是哈密顿算子在某个正定内积下自伴的充分条件,也不是必要条件. 相似文献
14.
For a large class of time-dependent non-Hermitian Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzed further and shown to be quasi-exactly solvable. Technically we constructed the Lewis–Riesenfeld invariants making use of the metric picture, which is an equivalent alternative to the Schrödinger, Heisenberg and interaction picture containing the time-dependence in the metric operator that relates the time-dependent Hermitian Hamiltonian to a static non-Hermitian Hamiltonian. 相似文献
15.
The topological phase transitions among normal insulator phase, two kinds of topological insulator phases, and topological semimetal phase are shown based on the non-Hermitian dimerized Su–Schrieffer–Heeger (SSH) model with the nonreciprocal intercell and long-range hopping. In contrast to the previous work, it is found that the topological insulator phase in the present SSH model can hold the larger non-Bloch winding number accompanied by exceptional winding of the generalized Brillouin zone around the gap-closing points. Compared with the usual topological insulator phase in non-Hermitian SSH model, the topological insulator with the larger winding number owns two pairs of zero energy modes with a distinct form of edge localization in the gap. The physical mechanism of the distinct edge localization for zero energy modes via a equivalent Hermitian version of the non-Hermitian SSH model is revealed. Additionally, the process of the phase transition is visualized among normal insulator phase, topological insulator phases, and topological semimetal phase in detail via the evolution of the gap-closing points on the plane of generalized Brillouin zone. This work further verifies the non-Bloch theory and enrich the investigation about the topologically nontrivial phase with the larger topological invariant in the non-Hermitian SSH model. 相似文献
16.
17.
研究相空间中离散力学系统对称性的摄动与绝热不变量.列出相空间中未受扰离散力学系统的特殊Lie对称性导致的Hojman型精确不变量.基于相空间中力学系统的高阶绝热不变量的定义,研究在小扰动作用下系统Lie对称性的摄动,得到了相空间中离散力学系统的一类新的绝热不变量——Hojman型绝热不变量.举例说明结果的应用.
关键词:
相空间
Lie对称性
摄动
绝热不变量 相似文献
18.
The concepts of spin and pseudospin symmetries has been used as mere rhetorics to decorate the pseudoscalar potential[Chin. Phys. B 22 090301(2013)]. It is also pointed out that a more complete analysis of the bound states of fermions in a pseudoscalar Cornell potential has already been published elsewhere. 相似文献
19.
Jing-Quan Li Jia-Jie Li Lu Qi Zhi-Xu Zhang Ji Cao Wen-Xue Cui Shou Zhang Hong-Fu Wang 《Annalen der Physik》2023,535(8):2300133
The symmetries and topological properties of the topological counterparts in 1D non-Hermitian systems are investigated. It is found that, after applying the non-unitary similarity transformation, the non-unitary topological counterpart in real space exhibits completely different global symmetries except for the sublattice symmetry and reveals many brand new local symmetries. Due to the abundant symmetries of non-unitary topological counterparts, it is also found that the unique overlapping projections about the unit sphere vector representing the eigenstates appear in the nontrivial regions, and the triviality of the point-gap topology of non-unitary topological counterpart completely eliminate the intrinsic skin effect in non-Hermitian systems. It is also shown that the unitary topological counterpart never arises any changes for the original symmetries and topological structures even in real space. Unitary topological counterparts are further summarized about the two-band Bloch Hamiltonian, which can expand the definition of non-Bloch winding number. Furthermore, it is demonstrated theoretically that the Bloch Hamiltonian would still hold time-reversal symmetry, abnormal particle-hole symmetry, and sublattice symmetry even suffering from the non-unitary transformation. This work provides a new way to understand the roles of symmetry and topology in non-Hermitian systems from the perspective of topological counterparts. 相似文献
20.
Shao-Ming Fei 《Czechoslovak Journal of Physics》2006,56(9):887-892
We study the quantum entanglement and separability of Hermitian and pseudo-Hermitian systems of identical bosonic or fermionic
particles with point interactions. The separability conditions are investigated in detail. 相似文献