\(L^ p\)-theory of Schrödinger semigroups. II. (English. Russian original) Zbl 0815.47050

Sib. Math. J. 31, No. 4, 540-549 (1990); translation from Sib. Mat. Zh. 31, No. 4(182), 16-26 (1990).
Let \(-\Delta +V\) be the Schrödinger operator with potential \(V\). Define three operators \(H_{p,\min}\), \(H_{p,\max}\), \(H_ p\) as \[ \begin{aligned} H_{p,\min} &=: [-\Delta+ V\restriction C_ 0^ \infty (\mathbb{R}^ d) ]^ \sim_{L^ p\to L^ p} \qquad (V\in L^ p_{\text{loc}} (\mathbb{R}^ d)),\\ H_{p,\max} f &=: -\Delta f+ Vf, \qquad f\in \text{Dom} (H_{p,\max}),\\ \text{Dom} (H_{p,\max}) &=: \{f\in L^ p: Vf\in L^ 1_{\text{loc}} (\mathbb{R}^ d) \text{ and } -\Delta f+ Vf\in L^ p \text{ in the distribution sense}\}, \end{aligned} \] and \(- H_ p\) is the generator of the bounded \(C_ 0\)-semigroup of type \(\omega\) \[ \varepsilon_ p (t)=: [e^{- tH} \restriction [L^ 2 \cap L^ p ]]^ \sim_{L^ p\to L^ p}, \qquad t>0. \] After proving some basic properties of \(H_ p\), the authors deduce many simple conditions under which either \(H_ p=H_{p,\max}\) or \(H_ p=H_{p,\min}\) holds. Their study is motivated by problems of T. Kato [Aspects of positivity in Funct. Anal., North-Holland Math. Stud. 122, 63–78 (1986; Zbl 0627.47025)], J. Voigt [J. Funct. Anal. 67, 167–205 (1986; Zbl 0628.47027)].
[For part I see V. F. Kovalenko and the second author, Ukr. Mat. Zh. 41, No. 2, 273–278 (1989; Zbl 0683.47025)].


47D06 One-parameter semigroups and linear evolution equations
47D03 Groups and semigroups of linear operators
35Q40 PDEs in connection with quantum mechanics
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