On the singular Weyl-Titchmarsh function of perturbed spherical Schrödinger operators. (English) Zbl 1219.34037
The authors consider perturbed spherical Schrödinger operators (also known as Bessel operators) of the form
\[ -\frac{d^2}{dx^2}+\frac{l(l+1)}{x^2}+q(x),\quad l\geq-\tfrac{1}{2},\;x\in(0,\infty), \]
where the potential \(q\) is real-valued satisfying \(q\in L_{\text{loc}}^1(0,\infty)\), \(xq(x)\in L^1(0,1)\) for \(l>-1/2\), and \(x(1-\log(x))q(x)\in L^1(0,1)\) for \(l=-1/2\). They investigate the singular Weyl-Titchmarsh \(m\)-function and show existence and detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this they show that the singular \(m\)-function belongs to the generalized Nevanlinna class and connect their results with the theory of super singular perturbations. The paper contains appendices devoted to Hardy type inequalities, generalized Navanlinna functions, and the theory of super singular perturbations.
\[ -\frac{d^2}{dx^2}+\frac{l(l+1)}{x^2}+q(x),\quad l\geq-\tfrac{1}{2},\;x\in(0,\infty), \]
where the potential \(q\) is real-valued satisfying \(q\in L_{\text{loc}}^1(0,\infty)\), \(xq(x)\in L^1(0,1)\) for \(l>-1/2\), and \(x(1-\log(x))q(x)\in L^1(0,1)\) for \(l=-1/2\). They investigate the singular Weyl-Titchmarsh \(m\)-function and show existence and detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this they show that the singular \(m\)-function belongs to the generalized Nevanlinna class and connect their results with the theory of super singular perturbations. The paper contains appendices devoted to Hardy type inequalities, generalized Navanlinna functions, and the theory of super singular perturbations.
Reviewer: Pavel Rehak (Brno)
MSC:
| 34B20 | Weyl theory and its generalizations for ordinary differential equations |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 34B30 | Special ordinary differential equations (Mathieu, Hill, Bessel, etc.) |
| 34L05 | General spectral theory of ordinary differential operators |
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