Geodesic
Eigenfunction expansions for the Schrödinger equation with an inverse-square potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 187 (2016) no. 2, pp. 360-382 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {A. G. Smirnov},
     title = {Eigenfunction expansions for {the~Schr\"odinger} equation with an~inverse-square potential},
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     pages = {360--382},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2016_187_2_a10/}
}
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A. G. Smirnov. Eigenfunction expansions for the Schrödinger equation with an inverse-square potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 187 (2016) no. 2, pp. 360-382. http://geodesic.mathdoc.fr/item/TMF_2016_187_2_a10/

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