# The resolvent for Laplace-type operators on asymptotically conic spaces

Andrew Hassell^{[1]}; András Vasy^{[2]}

- [1] Australian National University, Centre for Mathematics and its Applications, Canberra ACT 0200 (Australie)
- [2] Massachusetts Institute of Technology, Department of Mathematics, Cambridge MA (USA)

Annales de l’institut Fourier (2001)

- Volume: 51, Issue: 5, page 1299-1346
- ISSN: 0373-0956

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topHassell, Andrew, and Vasy, András. "The resolvent for Laplace-type operators on asymptotically conic spaces." Annales de l’institut Fourier 51.5 (2001): 1299-1346. <http://eudml.org/doc/115949>.

@article{Hassell2001,

abstract = {Let $X$ be a compact manifold with boundary, and $g$ a scattering metric on $X$, which
may be either of short range or “gravitational” long range type. Thus, $g$ gives $X$ the
geometric structure of a complete manifold with an asymptotically conic end. Let $H$ be
an operator of the form $H =\Delta + P$, where $\Delta $ is the Laplacian with respect to
$g$ and $P$ is a self-adjoint first order scattering differential operator with
coefficients vanishing at $\partial X$ and satisfying a “gravitational” condition. We
define a symbol calculus for Legendre distributions on manifolds with codimension two
corners and use it to give a direct construction of the resolvent kernel of $H$,
$R(\sigma + i0)$, for $\sigma $ on the positive real axis. In this approach, we do not use
the limiting absorption principle at any stage; instead we construct a parametrix which
solves the resolvent equation up to a compact error term and then use Fredholm theory to
remove the error term.},

affiliation = {Australian National University, Centre for Mathematics and its Applications, Canberra ACT 0200 (Australie); Massachusetts Institute of Technology, Department of Mathematics, Cambridge MA (USA)},

author = {Hassell, Andrew, Vasy, András},

journal = {Annales de l’institut Fourier},

keywords = {Legendre distributions; symbol calculus; scattering metrics; resolvent kernel},

language = {eng},

number = {5},

pages = {1299-1346},

publisher = {Association des Annales de l'Institut Fourier},

title = {The resolvent for Laplace-type operators on asymptotically conic spaces},

url = {http://eudml.org/doc/115949},

volume = {51},

year = {2001},

}

TY - JOUR

AU - Hassell, Andrew

AU - Vasy, András

TI - The resolvent for Laplace-type operators on asymptotically conic spaces

JO - Annales de l’institut Fourier

PY - 2001

PB - Association des Annales de l'Institut Fourier

VL - 51

IS - 5

SP - 1299

EP - 1346

AB - Let $X$ be a compact manifold with boundary, and $g$ a scattering metric on $X$, which
may be either of short range or “gravitational” long range type. Thus, $g$ gives $X$ the
geometric structure of a complete manifold with an asymptotically conic end. Let $H$ be
an operator of the form $H =\Delta + P$, where $\Delta $ is the Laplacian with respect to
$g$ and $P$ is a self-adjoint first order scattering differential operator with
coefficients vanishing at $\partial X$ and satisfying a “gravitational” condition. We
define a symbol calculus for Legendre distributions on manifolds with codimension two
corners and use it to give a direct construction of the resolvent kernel of $H$,
$R(\sigma + i0)$, for $\sigma $ on the positive real axis. In this approach, we do not use
the limiting absorption principle at any stage; instead we construct a parametrix which
solves the resolvent equation up to a compact error term and then use Fredholm theory to
remove the error term.

LA - eng

KW - Legendre distributions; symbol calculus; scattering metrics; resolvent kernel

UR - http://eudml.org/doc/115949

ER -

## References

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