On the threshold for Kato's selfadjointness problem and its L^p-generalization

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December 2014, 3(4): 699-711. doi: 10.3934/eect.2014.3.699

On the threshold for Kato's selfadjointness problem and its $L^p$-generalization

Motohiro Sobajima ^1,

1.	Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, Japan

Received March 2014 Revised May 2014 Published October 2014

Abstract
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In this paper the selfadjointness problem for Schrödinger operators $Au = -div(a\nabla u)+Vu$ in $\mathbb{R}^N$ $(N\in\mathbb{N})$ posed by Kato in [5] and its $L^p$-generalization ($1< p <\infty$) are dealt with. Under $|a(x)|\leq k(1+|x|)^{l+2}$ and $V(x)\geq c|x|^{l}$, the precise lower bounds of $c$ for (essential) selfadjointness in $L^2$ and $m$-sectoriality in $L^p$ of minimal and maximal realizations of $A$ are given. The proof is based on the method in Davies [1,Example 3.5]. This result is a (negative) answer to Kato's selfadjointness problem, and asserts that the lower bounds of $c$ stated in [7,Section 5] for $p=2$ and in [12,Section 3] for general $p$, are precise.

Keywords: selfadjointness, $m$-sectoriality, Schrödinger operators, Kato's selfadjointness problem..

Mathematics Subject Classification: Primary: 35J10; Secondary: 47B2.

Citation: Motohiro Sobajima. On the threshold for Kato's selfadjointness problem and its $L^p$-generalization. Evolution Equations and Control Theory, 2014, 3 (4) : 699-711. doi: 10.3934/eect.2014.3.699

References:

[1]	E. B. Davies, $L^{1}$ properties of second order elliptic operators, Bull. London Math. Soc., 17 (1985), 417-436. doi: 10.1112/blms/17.5.417.
[2]	A. Devinatz, Essential self-adjointness of Schrödinger-type operators, J. Functional Analysis, 25 (1977), 58-69. doi: 10.1016/0022-1236(77)90032-5.
[3]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Revised Third Printing, Springer-Verlag, Berlin, 1998.
[4]	D. M. Gitman, I. V. Tyutin and B. L. Voronov, Self-adjoint Extensions in Quantum Mechanics, General Theory and Applications to Schrodinger and Dirac Equations with Singular Potentials, Progress in Mathematical Physics 62, Birkhäuser/Springer, New York, 2012. doi: 10.1007/978-0-8176-4662-2.
[5]	T. Kato, Remarks on the selfadjointness and related problems for differential operators, Spectral theory of differential operators (Birmingham, Ala., 1981), Math. Stud., North-Holland, Amsterdam-New York, 55 (1981), 253-266. doi: 10.1016/S0304-0208(08)71641-4.
[6]	G. Metafune, N. Okazawa, M. Sobajima and C. Spina, Scale invariant elliptic operators with singular coefficients, preprint, arXiv:1405.5657.
[7]	G. Metafune, D. Pallara, P. J. Rabier and R. Schnaubelt, Uniqueness for elliptic operators on $L^p(\mathbb{R}^N)$ with unbounded coefficients, Forum Math., 22 (2010), 583-601. doi: 10.1515/forum.2010.031.
[8]	G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 303-340. doi: 10.2422/2036-2145.201010_012.
[9]	N. Okazawa, Sectorialness of second order elliptic operators in divergence form, Proc. Amer. Math. Soc., 113 (1991), 701-706. doi: 10.1090/S0002-9939-1991-1072347-4.
[10]	B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8.
[11]	M. Sobajima, $L^p$-theory for second-order elliptic operators with unbounded coefficients, J. Evol. Equ., 12 (2012), 957-971. doi: 10.1007/s00028-012-0163-1.
[12]	M. Sobajima, $L^p$-theory for second-order elliptic operators with unbounded coefficients in an endpoint class, J. Evol. Equ., 14 (2014), 461-475. doi: 10.1007/s00028-014-0223-9.

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References:

[1]	E. B. Davies, $L^{1}$ properties of second order elliptic operators, Bull. London Math. Soc., 17 (1985), 417-436. doi: 10.1112/blms/17.5.417.
[2]	A. Devinatz, Essential self-adjointness of Schrödinger-type operators, J. Functional Analysis, 25 (1977), 58-69. doi: 10.1016/0022-1236(77)90032-5.
[3]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Revised Third Printing, Springer-Verlag, Berlin, 1998.
[4]	D. M. Gitman, I. V. Tyutin and B. L. Voronov, Self-adjoint Extensions in Quantum Mechanics, General Theory and Applications to Schrodinger and Dirac Equations with Singular Potentials, Progress in Mathematical Physics 62, Birkhäuser/Springer, New York, 2012. doi: 10.1007/978-0-8176-4662-2.
[5]	T. Kato, Remarks on the selfadjointness and related problems for differential operators, Spectral theory of differential operators (Birmingham, Ala., 1981), Math. Stud., North-Holland, Amsterdam-New York, 55 (1981), 253-266. doi: 10.1016/S0304-0208(08)71641-4.
[6]	G. Metafune, N. Okazawa, M. Sobajima and C. Spina, Scale invariant elliptic operators with singular coefficients, preprint, arXiv:1405.5657.
[7]	G. Metafune, D. Pallara, P. J. Rabier and R. Schnaubelt, Uniqueness for elliptic operators on $L^p(\mathbb{R}^N)$ with unbounded coefficients, Forum Math., 22 (2010), 583-601. doi: 10.1515/forum.2010.031.
[8]	G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $L^p$ spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 303-340. doi: 10.2422/2036-2145.201010_012.
[9]	N. Okazawa, Sectorialness of second order elliptic operators in divergence form, Proc. Amer. Math. Soc., 113 (1991), 701-706. doi: 10.1090/S0002-9939-1991-1072347-4.
[10]	B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8.
[11]	M. Sobajima, $L^p$-theory for second-order elliptic operators with unbounded coefficients, J. Evol. Equ., 12 (2012), 957-971. doi: 10.1007/s00028-012-0163-1.
[12]	M. Sobajima, $L^p$-theory for second-order elliptic operators with unbounded coefficients in an endpoint class, J. Evol. Equ., 14 (2014), 461-475. doi: 10.1007/s00028-014-0223-9.

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