Reduction by symmetries in singular quantum-mechanical problems: General scheme and application to Aharonov-Bohm model

Abstract

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.

Eigenfunction expansions for the Schrödinger equation with an inverse-square potential

“…(6. 19) Indeed, let φ ∈ [0, π]. If α < 0, then π 2 α ≤ (π − φ) 2 α and, therefore, 0 < Sinc((π−φ) 2 α/4) ≤ Sinc(π 2 α/4) by Lemma B.1.…”

“…In [20], we considered problem (1.1) with α = κ 2 and constructed a parametrization of self-adjoint realizations of (1.2) and corresponding eigenfunction expansions that is continuous in κ on the interval (−1, 1) (and, in particular, at κ = 0). This work was motivated by our previous research [19] of the Aharonov-Bohm model, where zero and nonzero κ correspond to integer and noninteger values of the dimensionless magnetic flux through the solenoid. In terms of α, the results of [20] give a continuous transition from the region 0 < α < 1 to α = 0.…”

О Разложениях По Собственным Функциям Для Уравнения Шредингера С Обратноквадратичным Потенциалом

“…Такая ситуация не вполне удовлетворительна с физической точки зрения. Самосопряженные операторы, связанные с выражением (3), можно использовать, в том числе, для построения самосопряженных реализаций гамильтониана Ааронова-Бома [10]. При этом κ, равное нулю, и κ, отличные от нуля, отвечают соответственно целым и нецелым значениям безразмерного магнитного потока сквозь соленоид.…”

“…Использование таких сингулярных m-функций приводит к заметному упрощению при работе с разложениями по собственным функциям по сравнению с общей теорией [13], [5], основанной на матричнозначных мерах (отметим вместе 8) В данной статье термин "спектральная мера" всюду используется по отношению к определенным положительным мерам на R, точное определение которых будет дано ниже в предложении 14. Это отличается от терминологии, принятой в работе [10], где указанный термин применялся к проекторнозначным мерам в гильбертовом пространстве.…”

Eigenfunction expansions for the Schrödinger equation with inverse-square potential

“…This situation is not quite satisfactory from the physical standpoint. In particular, self-adjoint operators associated with (3) can be used to construct self-adjoint realizations of Aharonov-Bohm Hamiltonian [10], in which case zero and nonzero κ correspond to integer and noninteger values of the dimensionless magnetic flux through the solenoid. Hence, the existence of a welldefined limit κ → 0 is necessary to ensure the continuous transition between integer and noninteger values of the flux in the Aharonov-Bohm model.…”