Square Laplacian perturbed by inverse fourth-power potential. I Self-adjointness (real case)

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<jats:p>The self-adjointness of Δ<jats:sup>2</jats:sup> + κ|<jats:italic>x</jats:italic>|<jats:sup>−4</jats:sup> (κ>κ<jats:sub>0</jats:sub> = κ<jats:sub>0</jats:sub>(<jats:italic>N</jats:italic>)) in <jats:italic>L</jats:italic><jats:sup>2</jats:sup>(ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup>) is established as an application of the perturbation theorem in terms of Re(A<jats:italic>u, B<jats:sub>ε</jats:sub>u</jats:italic>), <jats:italic>u</jats:italic> ∈ <jats:italic>D(A</jats:italic>), for two non-negative self-adjoint operators <jats:italic>A, B</jats:italic> in a Hilbert space, where the family {<jats:italic>B</jats:italic><jats:sub>ε</jats:sub>}<jats:sub>ε>0</jats:sub> is the Yosida approximation of <jats:italic>B</jats:italic>. A key to the proof lies in a new inequality for the functions ν ∈ <jats:italic>L</jats:italic><jats:sup>2</jats:sup>(ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup>) with |x|<jats:sup>2</jats:sup>Δν ∈ <jats:italic>L</jats:italic><jats:sup>2</jats:sup>(ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup>) derived by using two real parameters.</jats:p>

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