文摘
We consider the operator$$L = - (d/dx)^{2}y(x) + x^{2} y + w(x) y \quad in L^{2}(\mathbb{R}),$$where$$w(x) = s \left[ \delta(x - b) - \delta(x + b) \right], \quad b \neq 0 \, \, real, \quad s \in \mathbb{C}.$$This operator has a discrete spectrum; eventually the eigenvalues are simple and$$\lambda_n = (2n + 1) + s^2\, \frac{\kappa(n)}{n} + \rho(n),$$ (0.1)where$$\kappa(n) = \frac{1}{2\pi}\left[(-1)^{n + 1} \sin \left( 2 b \sqrt{2n}\right) - \frac{1}{2} \sin \left( 4 b \sqrt{2n} \right) \right]$$ (0.2)and$$\vert \rho(n) \vert \leq C \frac{\log n}{n^{3/2}}. \label{eq:abstracterr}$$ (0.3)The analogue of (0.1)–(0.3) is given in the case of any two-point interaction perturbation$$w(x) = c_+ \delta{(x - b)} + c_- \delta{(x + b)}, \quad c_+, c_- \in \mathbb{C}.$$KeywordsHarmonic OscillatorHermite functionsAsymptotic of eigenvaluesMathematics Subject ClassificationPrimary 47E05Secondary 34L4034L15References1.Adduci, J., Mityagin, B.: Eigensystem of an L 2-perturbed harmonic oscillator is an unconditional basis. 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