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On the eigenproblems of PT-symmetric oscillators

K. C. Shin
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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.

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K. C. Shin. 2000-07-05. On the eigenproblems of PT-symmetric oscillators. https://doi.org/10.1063/1.1366328

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