This paper explains the systematics of the generation of families of spectra for the -symmetric quantum-mechanical Hamiltonians H=p(2)+x(2)(ix)(ε), H=p(2)+(x(2))(δ) and H=p(2)-(x(2))(μ). In addition, it contrasts the results obtained with those found for a bosonic scalar field theory, in particular in one dimension, highlighting the similarities to and differences from the quantum-mechanical case. It is shown that the number of families of spectra can be deduced from the number of non-contiguous pairs of Stokes wedges that display PT symmetry. To do so, simple arguments that use the Wentzel-Kramers-Brillouin approximation are used, and these imply that the eigenvalues are real. However, definitive results are in most cases presently only obtainable numerically, and not all eigenvalues in each family may be real. Within the approximations used, it is illustrated that the difference between the quantum-mechanical and the field-theoretical cases lies in the number of accessible regions in which the eigenfunctions decay exponentially. This paper reviews and implements well-known techniques in complex analysis and PT-symmetric quantum theory.
One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form
H=p2+V(x), where
V(x) is odd in
x, pure imaginary, and vanishes as
∣x∣→∞. Five PT-symmetric potentials are studied: the Scarf-II potential
V1(x)=iA1sech(x)tanh(x), which decays exponentially for large
∣x∣; the rational potentials
V2(x)=iA2x/(1+x4) and
V3(x)=iA3x/(1+∣x∣3), which decay algebraically for large
∣x∣; the step-function potential
V4(x)=iA4sgn(x)θ(2.5−∣x∣), which has compact support; the regulated Coulomb potential
V5(x)=iA5x/(1+x2), which decays slowly as
∣x∣→∞ and may be viewed as a long-range potential. The real parameters
An measure the strengths of these potentials. Numerical techniques for solving the time-independent Schrödinger eigenvalue problems associated with these potentials reveal that the spectra of the corresponding Hamiltonians exhibit universal properties. In general, the eigenvalues are partly real and partly complex. The real eigenvalues form the continuous part of the spectrum and the complex eigenvalues form the discrete part of the spectrum. The real eigenvalues range continuously in value from
0 to
+∞. The complex eigenvalues occur in discrete complex-conjugate pairs and for
Vn(x) (
1≤n≤4) the number of these pairs is finite and increases as the value of the strength parameter
An increases. However, for
V5(x) there is an {\it infinite} sequence of discrete eigenvalues with a limit point at the origin. This sequence is complex, but it is similar to the Balmer series for the hydrogen atom because it has inverse-square convergence.
This paper proposes to broaden the canonical formulation of quantum mechanics.Ordinarily, one imposes the condition H † = H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition.This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum.However, replacing this mathematical condition by the weaker and more physical requirement H ‡ = H, where ‡ represents combined parity reflection and time reversal PT , one obtains new classes of complex Hamiltonians whose spectra are still real and positive.This generalization of Hermiticity is investigated using a complex deformation H = p 2 + x 2 (ix) ǫ of the harmonic oscillator Hamiltonian, where ǫ is a real parameter.The system exhibits two phases: When ǫ ≥ 0, the energy spectrum of H is real and positive as a consequence of PT symmetry.However, when -1 < ǫ < 0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken.The phase transition that occurs at ǫ = 0 manifests itself in both the quantum-mechanical system and the underlying classical system.Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H = p 2 + x 2N (ix) ǫ with N integer and ǫ > -N ; each of these complex Hamiltonians exhibits a phase transition at ǫ = 0.These PT -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
We describe a method that allows for a practical application of the theory of pseudo-Hermitian operators to PT -symmetric systems defined on a complex contour. We apply this method to study the Hamiltonians H = p 2 + x 2 (i x ) ν with ν (−2, ∞ ) that are defined along the corresponding anti-Stokes lines. In particular, we reveal the intrinsic non-Hermiticity of H for the cases that ν is an even integer, so that H = p 2 ± x 2+ν , and give a proof of the discreteness of the spectrum of H for all ν (−2, ∞ ). Furthermore, we study the consequences of defining a square-well Hamiltonian on a wedge-shaped complex contour. This yields a PT -symmetric system with a finite number of real eigenvalues. We present a comprehensive analysis of this system within the framework of pseudo-Hermitian quantum mechanics. We also outline a direct pseudo-Hermitian treatment of PT -symmetric systems defined on a complex contour which clarifies the underlying mathematical structure of the formulation of PT -symmetric quantum mechanics based on the charge-conjugation operator. Our results provide conclusive evidence that pseudo-Hermitian quantum mechanics provides a complete description of general PT -symmetric systems regardless of whether they are defined along the real line or a complex contour.
It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this Review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.
We describe a method that allows for a practical application of the theory of pseudo-Hermitian operators to PT-symmetric systems defined on a complex contour. We apply this method to study the Hamiltonians
H=p2+x2(ix)ν with
ν∈(−2,∞) that are defined along the corresponding anti-Stokes lines. In particular, we reveal the intrinsic non-Hermiticity of
H for the cases that
ν is an even integer, so that
H=p2±x2+ν, and give a proof of the discreteness of the spectrum of
H for all
ν∈(−2,∞). Furthermore, we study the consequences of defining a square-well Hamiltonian on a wedge-shaped complex contour. This yields a PT-symmetric system with a finite number of real eigenvalues. We present a comprehensive analysis of this system within the framework of pseudo-Hermitian quantum mechanics. We also outline a direct pseudo-Hermitian treatment of PT-symmetric systems defined on a complex contour which clarifies the underlying mathematical structure of the formulation of PT-symmetric quantum mechanics based on the charge-conjugation operator. Our results provide a conclusive evidence that pseudo-Hermitian quantum mechanics provides a complete description of general PT-symmetric systems regardless of whether they are defined along the real line or a complex contour.
A major mathematical problem in PT-symmetric quantum mechanics is to determine whether or not the spectrum of any given non self-adjoint but PT-symmetric Schroedinger operator is real. In this case the PT-symmetry is called "proper". Clearly in this connection an equally important issue is the spontaneous violation of the PT-symmetry which might occur in a PT-symmetric operator family. The spontaneous violation of the PT-symmetry is defined as the transition from real values of the spectrum to complex ones at the variation of the parameter labeling the family. Its occurrence is referred to also as the
PT-symmetric phase transition. \nThis chapter intends to present a review of recent results concerning these two mathematical points, within the standard notions of spectral theory for Hilbert space operators. The main technical instrument is represented by perturbation theory.