受点相互作用扰动的谐振子算子的谱
谱理论
2015-06-22 v1 数学物理
math.MP
摘要
我们考虑算子 L = − ( d / d x ) 2 + x 2 y + w ( x ) y , y ∈ L 2 ( R ) L = - (d/dx)^2 + x^2 y + w(x) y , y \in L^2(\mathbb{R}) L = − ( d / d x ) 2 + x 2 y + w ( x ) y , y ∈ L 2 ( R ) w ( x ) = s [ δ ( x − b ) − δ ( x + b ) ] , b ≠ 0 , w(x) = s [ \delta(x - b) - \delta(x + b)], b \neq 0, w ( x ) = s [ δ ( x − b ) − δ ( x + b )] , b = 0 , s ∈ C s \in \mathbb{C} s ∈ C λ n = ( 2 n + 1 ) + s 2 ( κ ( n ) / n ) + ρ ( n ) \lambda_n = (2n + 1) + s^2 (\kappa(n) / n) + \rho(n) λ n = ( 2 n + 1 ) + s 2 ( κ ( n ) / n ) + ρ ( n ) κ ( n ) = 1 2 π [ ( − 1 ) n + 1 sin ( 2 b 2 n ) − 1 2 sin ( 4 b 2 n ) ] \kappa(n) = \frac{1}{2\pi} [(-1)^{n + 1} \sin ( 2 b \sqrt{2n} ) - \frac{1}{2} \sin ( 4 b \sqrt{2n} ) ] κ ( n ) = 2 π 1 [( − 1 ) n + 1 sin ( 2 b 2 n ) − 2 1 sin ( 4 b 2 n )] ∣ ρ ( n ) ∣ ≤ C ( log n ) / ( n 3 / 2 ) |\rho(n) | \leq C (\log n) / (n^{3/2}) ∣ ρ ( n ) ∣ ≤ C ( log n ) / ( n 3/2 ) s = i γ s = i \gamma s = iγ γ \gamma γ T ( γ ) T(\gamma) T ( γ ) T ( γ ) ≤ [ C ( 1 + ∣ γ ∣ ) log ( e + ∣ γ ∣ ) ] 2 . T(\gamma) \leq [ C (1 + | \gamma |) \log (e + | \gamma |)]^2. T ( γ ) ≤ [ C ( 1 + ∣ γ ∣ ) log ( e + ∣ γ ∣ ) ] 2 . w ( x ) = c + δ ( x − b ) + c − δ ( x + b ) , c + , c − ∈ C , w(x) = c_+ \delta(x - b) + c_- \delta(x + b), c_+, c_- \in \mathbb{C}, w ( x ) = c + δ ( x − b ) + c − δ ( x + b ) , c + , c − ∈ C ,
引用
@article{arxiv.1407.4153,
title = {The spectrum of a Harmonic Oscillator Operator Perturbed by Point Interactions},
author = {Boris Mityagin},
journal= {arXiv preprint arXiv:1407.4153},
year = {2015}
}
备注
65 pages