We prove the following. 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 (R 1 ) is discrete and eventually simple. Its SEAF (system of eigen-and associated functions) is an unconditional basis in L 2 (R).2000 Mathematics Subject Classification. 47E05, 34L40, 34L10.

Abstract. We analyze the perturbations T +B of a selfadjoint operator T in a Hilbert space H with discrete spectrum {t k } , T φ k = t k φ k , as an extension of our constructions in [1] where T was a harmonic oscillator operator. In particular, if t k+1 − t k ≥ ck α−1 , α > 1/2 and Bφ k = o(k α−1 ) then the system of root vectors of T + B, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in H. Statement of main resultsLet H be a separable Hilbert space. Consider an operator T with domain domT whose spectrum consists of a countable set of eigenvalues τ = {t k } ∞ k=1 with corresponding eigenvectors {φ k },which form an orthonormal basis in H. Let us also assume that t k+1 −t k > 0 and that for some fixed p ∈ Z + , d > 0(1.1) Define △t k = t k+1 −t k . Then (1

Abstract. 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 En(z), En(0) = n 2 , is a well-defined analytic function. Let Rn be the convergence radius of its Taylor's series about z = 0. It is proved that Rn ≤ C(α)n 2−α if 0 ≤ α < 11/6.

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The solution of differential equations using the software package Mathematica is discussed in this paper. We focus on two functions, DSolve and NDSolve, and give various examples of how one can obtain symbolic or numerical results using these functions. An overview of the Wolfram Demonstrations Project is given, along with various novel user-contributed examples in the field of differential equations. The use of these Demonstrations in a classroom setting is elaborated upon to emphasize their significance for education.