| Title:
|
On $m$-sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry (English) |
| Author:
|
Milatovic, Ognjen |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
45 |
| Issue:
|
1 |
| Year:
|
2004 |
| Pages:
|
91-100 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider a Schrödinger-type differential expression $H_V=\nabla^*\nabla+V$, where $\nabla $ is a $C^{\infty}$-bounded Hermitian connection on a Hermitian vector bundle $E$ of bounded geometry over a manifold of bounded geometry $(M,g)$ with metric $g$ and positive $C^{\infty}$-bounded measure $d\mu$, and $V$ is a locally integrable section of the bundle of endomorphisms of $E$. We give a sufficient condition for $m$-sectoriality of a realization of $H_V$ in $L^2(E)$. In the proof we use generalized Kato's inequality as well as a result on the positivity of $u\in L^2(M)$ satisfying the equation $(\Delta _M+b)u=\nu $, where $\Delta _M$ is the scalar Laplacian on $M$, $b>0$ is a constant and $\nu\geq 0$ is a positive distribution on $M$. (English) |
| Keyword:
|
Schrödinger operator |
| Keyword:
|
$m$-sectorial |
| Keyword:
|
manifold |
| Keyword:
|
bounded geometry |
| Keyword:
|
singular potential |
| MSC:
|
35J10 |
| MSC:
|
35P05 |
| MSC:
|
47B25 |
| MSC:
|
58J05 |
| MSC:
|
58J50 |
| MSC:
|
81Q10 |
| idZBL:
|
Zbl 1127.35348 |
| idMR:
|
MR2076861 |
| . |
| Date available:
|
2009-05-05T16:43:26Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119438 |
| . |
| Reference:
|
[1] Aubin T.: Some Nonlinear Problems in Riemannian Geometry.Springer-Verlag, Berlin, 1998. Zbl 0896.53003, MR 1636569 |
| Reference:
|
[2] Braverman M., Milatovic O., Shubin M.: Essential self-adjointness of Schrödinger type operators on manifolds.Russian Math. Surveys 57 4 (2002), 641-692. Zbl 1052.58027, MR 1942115 |
| Reference:
|
[3] Kato T.: Schrödinger operators with singular potentials.Israel J. Math. 13 (1972), 135-148. MR 0333833 |
| Reference:
|
[4] Kato T.: A second look at the essential selfadjointness of the Schrödinger operators.Physical reality and mathematical description, Reidel, Dordrecht, 1974, pp.193-201. MR 0477431 |
| Reference:
|
[5] Kato T.: On some Schrödinger operators with a singular complex potential.Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 5 (1978), 105-114. Zbl 0376.47021, MR 0492961 |
| Reference:
|
[6] Kato T.: Perturbation Theory for Linear Operators.Springer-Verlag, New York, 1980. Zbl 0836.47009 |
| Reference:
|
[7] Milatovic O.: Self-adjointness of Schrödinger-type operators with singular potentials on manifolds of bounded geometry.Electron. J. Differential Equations, No. 64 (2003), 8pp (electronic). Zbl 1037.58013, MR 1993772 |
| Reference:
|
[8] Reed M., Simon B.: Methods of Modern Mathematical Physics I, II: Functional Analysis. Fourier Analysis, Self-adjointness.Academic Press, New York e.a., 1975. MR 0751959 |
| Reference:
|
[9] Shubin M.A.: Spectral theory of elliptic operators on noncompact manifolds.Astérisque no. 207 (1992), 35-108. MR 1205177 |
| Reference:
|
[10] Taylor M.: Partial Differential Equations II: Qualitative Studies of Linear Equations.Springer-Verlag, New York e.a., 1996. MR 1395149 |
| . |
