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3 papers
Eigenfunction expansions for the Schrödinger equation with an inverse-square potential
A. G. Smirnov Lebedev Physical Institute, RAS,
Moscow, Russia
Abstract:
We consider the one-dimensional Schrödinger equation
$-f''+q_\kappa f=Ef$ on the positive half-axis with the potential
$q_\kappa(r)=(\kappa^2-1/4) r^{-2}$. For each complex number
$\vartheta$, we construct a solution
$u^\kappa_\vartheta(E)$ of this equation that is analytic in
$\kappa$ in a complex neighborhood of the interval
$(-1,1)$ and, in particular, at the “singular” point
$\kappa=0$. For
$-1<\kappa<1$ and real
$\vartheta$, the solutions
$u^\kappa_\vartheta(E)$ determine a unitary eigenfunction expansion operator $U_{\kappa,\vartheta}\colon L_2(0,\infty)\to L_2(\mathbb R,\mathcal V_{\kappa,\vartheta})$, where
$\mathcal V_{\kappa,\vartheta}$ is a positive measure on
$\mathbb R$. We show that every self-adjoint realization of the formal differential expression
$-\partial^2_r+ q_\kappa(r)$ for the Hamiltonian is diagonalized by the operator
$U_{\kappa,\vartheta}$ for some
$\vartheta\in\mathbb R$. Using suitable singular Titchmarsh–Weyl
$m$-functions, we explicitly find the measures
$\mathcal V_{\kappa,\vartheta}$ and prove their continuity in
$\kappa$ and
$\vartheta$.
Keywords:
Schrödinger equation, inverse-square potential, self-adjoint extension,
eigenfunction expansion, Titchmarsh–Weyl $m$-function. Received: 24.08.2015
Revised: 09.11.2015
DOI:
10.4213/tmf9032