## The form sum and the Friedrichs extension of Schrödinger-type operators on Riemannian manifolds

HTML articles powered by AMS MathViewer

- by Ognjen Milatovic PDF
- Proc. Amer. Math. Soc.
**132**(2004), 147-156 Request permission

## Abstract:

We consider $H_V=\Delta _M+V$, where $(M,g)$ is a Riemannian manifold (not necessarily complete), and $\Delta _M$ is the scalar Laplacian on $M$. We assume that $V=V_0+V_1$, where $V_0\in L_{\operatorname {loc}}^2(M)$ and $-C\leq V_1\in L_{\operatorname {loc}}^1(M)$ ($C$ is a constant) are real-valued, and $\Delta _M+V_0$ is semibounded below on $C_{c}^{\infty }(M)$. Let $T_0$ be the Friedrichs extension of $(\Delta _M+V_0)|_{C_{c}^{\infty }(M)}$. We prove that the form sum $T_0\tilde {+} V_1$ coincides with the self-adjoint operator $T_F$ associated to the closure of the restriction to $C_{c}^{\infty }(M)\times C_{c}^{\infty }(M)$ of the sum of two closed quadratic forms of $T_0$ and $V_1$. This is an extension of a result of Cycon. The proof adopts the scheme of Cycon, but requires the use of a more general version of Kato’s inequality for operators on Riemannian manifolds.## References

- M. Braverman, O. Milatovic, M. A. Shubin,
*Essential self-adjointness of Schrödinger type operators on manifolds*, Russian Math. Surveys, 57 (4) (2002), 641–692. - Hans L. Cycon,
*On the form sum and the Friedrichs extension of Schrödinger operators with singular potentials*, J. Operator Theory**6**(1981), no. 1, 75–86. MR**637002** - William G. Faris,
*Self-adjoint operators*, Lecture Notes in Mathematics, Vol. 433, Springer-Verlag, Berlin-New York, 1975. MR**0467348** - David Gilbarg and Neil S. Trudinger,
*Elliptic partial differential equations of second order*, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR**0473443** - Heinz-Willi Goelden,
*On non-degeneracy of the ground state of Schrödinger operators*, Math. Z.**155**(1977), no. 3, 239–247. MR**609535**, DOI 10.1007/BF02028443 - H. Hess, R. Schrader, and D. A. Uhlenbrock,
*Domination of semigroups and generalization of Kato’s inequality*, Duke Math. J.**44**(1977), no. 4, 893–904. MR**458243** - H. Hess, R. Schrader, and D. A. Uhlenbrock,
*Kato’s inequality and the spectral distribution of Laplacians on compact Riemannian manifolds*, J. Differential Geometry**15**(1980), no. 1, 27–37 (1981). MR**602436** - Tosio Kato,
*Perturbation theory for linear operators*, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR**1335452** - Tosio Kato,
*A second look at the essential selfadjointness of the Schrödinger operators*, Physical reality and mathematical description, Reidel, Dordrecht, 1974, pp. 193–201. MR**0477431** - Leo F. Epstein,
*A function related to the series for $e^{e^x}$*, J. Math. Phys. Mass. Inst. Tech.**18**(1939), 153–173. MR**58**, DOI 10.1002/sapm1939181153 - Barry Simon,
*An abstract Kato’s inequality for generators of positivity preserving semigroups*, Indiana Univ. Math. J.**26**(1977), no. 6, 1067–1073. MR**461209**, DOI 10.1512/iumj.1977.26.26086 - Barry Simon,
*Kato’s inequality and the comparison of semigroups*, J. Functional Analysis**32**(1979), no. 1, 97–101. MR**533221**, DOI 10.1016/0022-1236(79)90079-X - Barry Simon,
*Maximal and minimal Schrödinger forms*, J. Operator Theory**1**(1979), no. 1, 37–47. MR**526289** - Michael E. Taylor,
*Partial differential equations. II*, Applied Mathematical Sciences, vol. 116, Springer-Verlag, New York, 1996. Qualitative studies of linear equations. MR**1395149**, DOI 10.1007/978-1-4757-4187-2

## Additional Information

**Ognjen Milatovic**- Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
- Address at time of publication: Department of Mathematics, Fitchburg State College, Fitchburg, Massachusetts 01420
- MR Author ID: 705360
- Email: ogmilato@lynx.neu.edu, omilatovic@fsc.edu
- Received by editor(s): August 20, 2002
- Published electronically: April 24, 2003
- Communicated by: David S. Tartakoff
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**132**(2004), 147-156 - MSC (2000): Primary 35P05, 58G25; Secondary 47B25, 81Q10
- DOI: https://doi.org/10.1090/S0002-9939-03-07075-8
- MathSciNet review: 2021257