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1. Notations. We use standard notations and basic facts from the theory of self-adjoint operators on Hilbert spaces. Let H be a separable Hilbert space with an inner product ( , ) and let A be a linear operator in H with the domain D(A) ⊂ H — a linear subset of H . , D(A) = H , the adjoint operator A∗ is an operator with the domain D(A∗ ) = { ϕ ∈ H | ∃ η ∈ H such that (Aψ, ϕ) = (ψ, η) ∀ ψ ∈ D(A)}, defined by η = A∗ ϕ. Operator A is called symmetric if (Aϕ, ψ) = (ϕ, Aψ) for all ϕ, ψ ∈ D(A). The regular set of a closed operator A with a dense domain D(A) is the set ρ(A) = {λ ∈ C | A−λI : D(A) → H is a bijection with a bounded inverse2}.

We summarize everything in the following statement. 1. Hamiltonian 2 d2 ω2q2 H=− + 2 dq 2 2 of harmonic oscillator, defined on S (R), is essentially self-adjoint operator on L2 (R) with pure point spectrum. The complete system of eigenfunctions is ωq 2 − ω ω 1 √ e 2 Hn ψn (q) = 4 q , π 2n n! where Hn (q) are classical Hermite polynomials, and Hψn = ω (n + 12 ). ¯ of H is self-adjoint and D(H) ¯ = W 2,2 (R) ∩ W 2,2 (R). The closure H Proof. The operator H is symmetric and has a complete system of eigenvectors, so that Im(H + iI) = Im(H − iI) = H .

Admissible classes of potential functions v(q) are determined by the requirement that H is (essentially) self-adjoint. In coordinate representation H has the standard form 2 ∆ + V, 2m where ∆ is the Laplacian of the standard Eucildean metric on Rn , H= n ∆=− k=1 ∂2 , ∂qk2 and V = v(q) is a multiplication by v(q) operator. 3. 1. Harmonic oscillator. In classical mechanics the simplest system with one degree of freedom is harmonic oscillator. Its phase space M = R2 carries canonical Poisson bracket and corresponding Hamiltonian function is p2 mω 2 q 2 h(p, q) = + , 2m 2 where ω > 0 has a physical meaning of frequency of the oscillations.

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