Regularization and convergence for singular perturbations
The authors prove a theorem in an abstract setting which is applicable to the determination of selfadjoint extensions of Schrödinger's operator when the potential is singular. Let \(A\) be a selfadjoint operator on Hilbert space \({\mathfrak h}\) and \(V\) a Hermitian operator such that for \(D\) dense in \((\text{dmn }A\cap\text{dmn }V)\), \((Vf,f)\leq a(Af,f)+ b(f,f)\), \(a<1\), \(b<\infty\), \(f\in D\). Define \(\{V_ n\}^ \infty_{n=1}\) to be a sequence of Hermitian operators \(\text{dmn }V_ n\supseteq\text{dmn }V\), \(H_ n= A-V_ n\), and let \(\gamma_ n\), \(\gamma\), \(\nu\) be symmetric forms \(\gamma_ n(f,g)= (V_ n f,g)\), \(\gamma(f,g)=(Vf,g)\), \(f,g\in D\), \(\nu(f,g)= (Af,g)\), \(f,g\in\text{dmn }A\). \(\widehat\nu(f,g)\) will be the closure of the symmetric form \(\nu_ 0(f,g)= \nu(f,g)\), \(f,g\in D\), and \(\widehat\gamma\) the extension of \(\gamma\) to \(\text{dmn}(\widehat\nu)\). The sequence \(\{V_ n\}\) is said to be a \(\widehat\nu\)-admissible regularizing sequence for \(V\) if (i) \(0\leq \gamma_ 1\leq\cdots\leq \gamma_ n\leq\cdots\leq \gamma\), (ii) \(\lim_{n\to\infty} \gamma_ n(f,f)= \gamma(f,f)\), \(f\in D\), (iii) \(\sup \gamma_ n (f,f)= \infty\) for all \(f\in M\), \(f\neq 0\), where \(M=\{h\mid h\in\text{dmn }\nu\), \(\nu(h,k)+ (h,k)= 0\), \(k\in\text{dmn}(\widehat\nu)\}\). It is proved that if \(\{V_ n\}\) is a \(\widehat\nu\)-admissible regularizing sequence for \(V\) and (iv) there exists an open set \(U\) and number \(\lambda\) such that \(U\subset(a,\infty)\), \(1\in U\), and \(\lambda<0\), \(\lambda<\inf(\widehat\nu- k^{-1}\widehat\gamma)\) for all \(k\in U\), \(\lambda\in\rho(H_ n)\), then s-\(\lim_{n\to\infty} (H_ n- z)^{-1}= (H- z)^{-1}\), \(z\in\{\lambda\}\cup (C^ 1\backslash R^ 1)\), defines a selfadjoint extension of \(A+ V\). If \(1\not\in U\) in (iv) then the results apply to \((A- k^{-1} V)\). Sufficient conditions are given for (iv) to hold. The theorem is applicable to determining selfadjoint extensions to the Schrödinger operator \((-\Delta)- k^{-1} V(x)\) corresponding to the singular potentials \(V(x)= {1\over 4}| x|^{-\alpha}\), \(0\leq \alpha\leq 2\) with \(k=1\) when \(\alpha<2\) and \(k>1\), \(\alpha=2\), \(0<x < \infty\).
determination of selfadjoint extensions of Schrödinger's operator
scientific article; zbMATH DE number 97874