Maximal and minimal forms for generalized Schrödinger operator (Q2926586)

In this paper, the authors study the form-core problem for generalized higher-order Schrödinger operators \(H= (-\Delta)^m + V\) with nonnegative potentials \(V\) on \(\mathbb{R}^n\). They consider the maximal closed form NEWLINE\[NEWLINE Q_{\max}(f,g) = \int_{\mathbb{R}^n} (-\Delta)^{m/2} f \cdot \overline{(-\Delta)^{m/2}g} \,{d}x + \int_{\mathbb{R}^n} V^{1/2} f \cdot \overline{V^{1/2}g} \,{d}x NEWLINE\]NEWLINE on the domain NEWLINE\[NEWLINE D(Q_{{\max}}) = \{ f\in W^{m,2}(\mathbb{R}^n) \; | \; V^{1/2}f \in L^2(\mathbb{R}^n) \} , NEWLINE\]NEWLINE where \(W^{m,2}(\mathbb{R}^n)\) denotes the Sobolev space of order \(m\), and prove that, under suitable local integrability conditions on \(V\), the maximal form \(Q_{{\max}}\) coincides with the form closure of its restriction to \(C^\infty_c (\mathbb{R}^n)\), that is, the minimal form \(Q_{{\min}}\). More precisely, the main result states that \(Q_{{\max}}\) and \(Q_{{\min}}\) coincide as soon as the nonnegative potential \(V\) is in \(L^p_{\mathrm{loc}}(\mathbb{R}^n)\), where NEWLINE\[NEWLINE p = n/(2m), \; {\text{if}} \; n > 2m; \quad p > 1, \; {\text{if}} \; n = 2m; \quad p = 1, \; {\text{if}} \; n < 2m . NEWLINE\]NEWLINE In particular, every continuous potential satisfies this condition.NEWLINENEWLINEThe second-order case \(m=1\) in the generality of \(0 \leq V \in L^1_{\mathrm {loc}}(\mathbb{R}^n)\) is a classical result of \textit{T. Kato} [Integral Equations Oper. Theory 1, 103--113 (1978; Zbl 0395.47023)]. However, as the authors point out, for \(m\geq 2\) the heat semigroup \(e^{-t(-\Delta)^m}\) is not positivity preserving, and it becomes difficult to use probabilistic tools based on the Feynman-Kac formula when the kernel of \(e^{-tH}\) is nonpositive. Instead, the proof of the main result is based on analyticity of \(e^{-tH}\) and the use of Sobolev embedding.NEWLINENEWLINEAs an application, a known upper bound for the kernel of \(e^{-t(-\Delta)^m}\) on \(\mathbb{R}^n\) with \(n < 2m\) is shown to persist in the presence of a nonnegative locally integrable potential.