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We consider the one-dimensional Schrödinger equation −f + qαf = Ef on the positive half-axis with the potential qα(r) = (α − 1/4)r −2 . It is known that the value α = 0 plays a special role in this problem: all self-adjoint realizations of the formal differential expression −∂ 2 r + qα(r) for the Hamiltonian have infinitely many eigenvalues for α < 0 and at most one eigenvalue for α ≥ 0. For each complex number ϑ, we construct a solution U α ϑ (E) of this equation that is entire analytic in α and, in particular, is not singular at α = 0. For α < 1 and real ϑ, the solutions U α ϑ (E) determine a unitary eigenfunction expansion operator U α,ϑ : L 2 (0, ∞) → L 2 (R, V α,ϑ ), where V α,ϑ is a positive measure on R. We show that each operator U α,ϑ diagonalizes a certain self-adjoint realization h α,ϑ of the expression −∂ 2 r + qα(r) and, moreover, that every such realization is equal to h α,ϑ for some ϑ ∈ R. Employing suitable singular Titchmarsh-Weyl m-functions, we explicitly find the spectral measures V κ,ϑ and prove their smooth dependence on α and ϑ. Using the formulas for the spectral measures, we analyse in detail how the transition through the point α = 0 occurs for both the eigenvalues and the continuous spectrum of h α,ϑ .

С точностью до преобразований эквивалентности найдены все непрерывные формальные деформации алгебры Пуассона, реализованной на гладких грас-сманозначных функциях с компактным носителем на R 2n при 2n 4. Показа-но, что у рассматриваемых алгебр существуют дополнительные деформации, отличные от скобки Мойала.

We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering twoand three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general scheme is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid. We construct all self-adjoint extensions of this Hamiltonian, invariant under translations along the solenoid and rotations around it, and explicitly find their eigenfunction expansions.Given a positive σ-finite measure ν, we say that a ν-a.e. defined family S of Hilbert spaces is νnondegenerate if S(s) = {0} for ν-a.e. s.Definition IV.1. Let X be a set of closed densely defined operators in a Hilbert space H. A triple (ν, S, V ), where ν is a positive σ-finite measure, S is a ν-nondegenerate ν-measurable family of Hilbert spaces, and V is a unitary operator from H to ⊕ S(s) dν(s), is called a diagonalization for X if every T ∈ X is equal to V −1 T ν,S g V for some complex ν-measurable function g. A diagonalization (ν, S, V ) for X is called exact if condition (5) holds for any complex ν-measurable ν-essentially bounded function g.

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