For any complex valued L p -function b(x), 2 ≤ p < ∞, or L ∞-function with the norm ‖b↾L ∞‖ < 1, the spectrum of a perturbed harmonic oscillator operator L = −d 2/dx 2 + x 2 + b(x) in L 2(ℝ1) is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in L 2(ℝ).

We analyze the perturbations $T+B$ of a selfadjoint operator $T$ in a Hilbert space $H$ with discrete spectrum $\{t_k \}$, $T \phi_k = t_k \phi_k$, as an extension of our constructions in arXiv: 0912.2722 where $T$ was a harmonic oscillator operator. In particular, if $t_{k+1}-t_k \geq c k^{\alpha - 1}, \quad \alpha > 1/2$ and $\| B \phi_k \| = o(k^{\alpha - 1})$ then the system of root vectors of $T+B$, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in $H$.

Consider a family of infinite tri-diagonal matrices of the form L + zB , where the matrix L is diagonal with entries L kk = k 2 , and the matrix B is off-diagonal, with nonzero entries B k , k +1 = B k +1, k = k α , 0 ≤ α < 2. The spectrum of L + zB is discrete. For small | z | the n th eigenvalue E n ( z ), E n (0) = n 2 , is a well-defined analytic function. Let R n be the convergence radius of its Taylor’s series about z = 0. It is proved that R_n ≤ C(α) n^2-α if 0 ≤α <11 /6 .