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\(L^ p\)-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)].

MSC:

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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[1] V. F. Kovalenko and Yu. A. Semenov, ?On the Lp-theory of Schrödinger semigroups. I,? Ukr. Mat. Zh.,41, No. 2, 273-278 (1989). · Zbl 0701.34073
[2] J. Voigt, ?Absorption semigroups, their generators, and Schrödinger semigroups,? J. Funct. Anal.,67, No. 2, 167-205 (1986). · Zbl 0628.47027
[3] Yu. A. Semenov, ?The essential self-adjointness of the operator-?+V,? Manuscript deposited at UkrNIINTI, January 4, 1987, No. 78-Uk87.
[4] T. Kato, ?A second look at the essential self-adjointness of the Schrödinger operators,? in: Physical Reality and Mathematical Description, C. P. Enz and J. Mehra (eds.), Reidel, Dordrecht, Holland (1974). · Zbl 0328.47023
[5] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York (1975). · Zbl 0308.47002
[6] V. F. Kovalenko, M. A. Perelmuter, and Yu. A. Semenov, ?Schrödinger operators with Lw 1/2(R1)-potentials,? J. Math. Phys.,22, No. 5, 1033-1044 (1981). · Zbl 0463.47027
[7] V. F. Kovalenko and Yu. A. Semenov, ?Semigroups generated by a second-order elliptic operator,? in Application of Methods of Functional Analysis in Problems of Mathematical Physics [in Russian], Akad. Nauk Ukr. SSR, Inst. Mat. Kiev (1987), pp. 17-36.
[8] V. F. Kovalenko and Yu. A. Semenov, ?The Lp-contractibility of the semigroup, generated by the Schrödinger operator with an unbounded negative potential,? Manuscript deposited at UkrNIINTI, September 30, 1985, No. 2381 Uk-85.
[9] T. Kato, ?Lp-theory of Schrödinger operators with a singular potential,? in: Aspects of Positivity in Functional Analysis (Tübingen, 1985), North-Holland, Amsterdam (1986). pp. 63-78.
[10] G. Talenti, ?Elliptic equations and rearrangements,? Ann. Scuola Norm. Sup. Pisa,3, No. 4, 697-718 (1976). · Zbl 0341.35031
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