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On the threshold for Kato's selfadjointness problem and its $L^p$-generalization
1. | Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, Japan |
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. |
show all references
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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