On the eigenproblems of PT-symmetric oscillators

Abstract

We consider the non-Hermitian Hamiltonian H= -\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a polynomial of degree at most n \geq 1 with all nonnegative real coefficients (possibly P\equiv 0). It is proved that the eigenvalues λmust be in the sector | arg λ| \leq \fracπ{2n+3}. Also for the case H=-\frac{d^2}{dx^2}-(ix)^3, we establish a zero-free region of the eigenfunction u and its derivative u^\prime and we find some other interesting properties of eigenfunctions.

21pages, 9 figures

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Cited by in corpus (25)

• Making Sense of Non-Hermitian Hamiltonians
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Pseudo-Supersymmetric Quantum Mechanics and Isospectral Pseudo-Hermi tian Hamiltonians
• Space of State Vectors in PT Symmetrical Quantum Mechanics
• Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics
• Biorthogonal Quantum Systems
• Bound states of the Dirac equation for the PT-symmetric generalized Hulthen potential by the Nikiforov-Uvarov method
• Spontaneous breakdown of PT symmetry in the solvable square-well model
• Exponential Type Complex and non-Hermitian Potentials in PT-Symmetric Quantum Mechanics
• Finite-Dimensional PT-Symmetric Hamiltonians
• PT symmetric pseudo-perturbation recipe; an imaginary cubic oscillator with spikes
• Solvability and PT-symmetry in a double-well model with point interactions
• Coupled-channel version of PT-symmetric square well
• All Hermitian Hamiltonians Have Parity
• Numerical Simulations of PT-Symmetric Quantum Field Theories
• Dirac and Klein-Gordon particles in complex Coulombic fields; a similarity transformation
• Solvable simulation of a double-well problem in PT symmetric quantum mechanics
• Two-Point Green's Function in PT-Symmetric Theories
• Eigenvalues of PT-symmetric oscillators with polynomial potentials
• The potential (iz)^m generates real eigenvalues only, under symmetric rapid decay conditions
• Supersymmetric Biorthogonal Quantum Systems
• Non-isospectrality of the generalized Swanson Hamiltonian and harmonic oscillator
• An introductory review on resource theories of generalized nonclassical light
• Ad hoc physical Hilbert spaces in Quantum Mechanics
• Comment on "Numerical estimates of the spectrum for anharmonic PT symmetric potentials" by Bowen et al