Anharmonic Oscillators in the Complex Plane, $boldsymbol{mathcal{PT}}$-symmetry,and Real Eigenvalues |
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Authors: | Kwang C. Shin |
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Affiliation: | 1.Department of Mathematics,University of West Georgia,Carrollton,USA |
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Abstract: | For integers m ≥ 3 and 1 ≤ ℓ ≤ m − 1, we study the eigenvalue problems − u ″(z) + [( − 1)ℓ(iz) m − P(iz)]u(z) = λu(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays argz=-fracp2±frac(l+1)pm+2arg z=-frac{pi}{2}pm frac{(ell+1)pi}{m+2} in the complex plane, where P is a polynomial of degree at most m − 1. We provide asymptotic expansions of the eigenvalues λ n . Then we show that if the eigenvalue problem is PTmathcal{PT}-symmetric, then the eigenvalues are all real and positive with at most finitely many exceptions. Moreover, we show that when gcd(m,l)=1gcd(m,ell)=1, the eigenvalue problem has infinitely many real eigenvalues if and only if one of its translations or itself is PTmathcal{PT}-symmetric. Also, we will prove some other interesting direct and inverse spectral results. |
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