具有屏蔽库仑核的谐振子阱是拟精确可解的
Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable
摘要：
在将哈密顿量的厄米性减弱到仅仅是PT不变性的量子化方案中，谐波势和库仑势的叠加V（x）= x^2+ Ze^2/x被定义为纯虚有效电荷（Ze^2=if），并通过x的纯虚位移正则化。这个模型是准精确可解的：我们证明了在每个激发态，（N+1）阶谐振子能量E=2N+3处，不仅存在已知的谐振子束缚态（在消失电荷f=0处），而且还存在进一步的基本Sturmian本征态的（N+1）-plet，在本征电荷f=f_n > 0，n = 0，1，. N.除了前几个最小的多重数N，我们建议他们的微扰建设。
英文摘要：
In the quantization scheme which weakens the hermiticity of a Hamiltonian to its mere PT invariance the superposition V(x) = x^2+ Ze^2/x of the harmonic and Coulomb potentials is defined at the purely imaginary effective charges (Ze^2=if) and regularized by a purely imaginary shift of x. This model is quasi-exactly solvable: We show that at each excited, (N+1)-st harmonic-oscillator energy E=2N+3 there exists not only the well known harmonic oscillator bound state (at the vanishing charge f=0) but also a normalizable (N+1)-plet of the further elementary Sturmian eigenstates ψ_n(x) at eigencharges f=f_n > 0, n = 0, 1, ..., N. Beyond the first few smallest multiplicities N we recommend their perturbative construction.
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