Unitary one-parameter groups with finite speed of propagation

Author:
E. C. Svendsen

Journal:
Proc. Amer. Math. Soc. **84** (1982), 357-361

MSC:
Primary 58G11; Secondary 35L40, 47D10

DOI:
https://doi.org/10.1090/S0002-9939-1982-0640230-6

MathSciNet review:
640230

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that is a Hermitian vector bundle over a Riemannian manifold and that is a one-parameter group of linear operators on the set of smooth sections of with compact support. We prove that if satisfies a smoothness condition, is unitary, and propagates initial data with finite speed, then it can be constructed from the solutions of a first-order symmetric hyperbolic system of partial differential equations.

**[1]**Stephen J. Berman,*Wave equations with finite velocity of propagation*, Trans. Amer. Math. Soc.**188**(1974), 127–148. MR**0330781**, https://doi.org/10.1090/S0002-9947-1974-0330781-6**[2]**Paul R. Chernoff,*Essential self-adjointness of powers of generators of hyperbolic equations*, J. Functional Analysis**12**(1973), 401–414. MR**0369890****[3]**I. M. Gel′fand and G. E. Šilov,*Fourier transforms of rapidly increasing functions and questions of the uniqueness of the solution of Cauchy’s problem*, Amer. Math. Soc. Transl. (2)**5**(1957), 221–274. MR**0083624****[4]**SigurÄ‘ur Helgason,*Differential geometry and symmetric spaces*, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR**0145455****[5]**Jaak Peetre,*Réctification à l’article “Une caractérisation abstraite des opérateurs différentiels”*, Math. Scand.**8**(1960), 116–120 (French). MR**0124611**, https://doi.org/10.7146/math.scand.a-10598

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
58G11,
35L40,
47D10

Retrieve articles in all journals with MSC: 58G11, 35L40, 47D10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1982-0640230-6

Keywords:
Symmetric hyperbolic system,
finite speed of propagation,
initial-value problem,
unitary one-parameter group,
Hermitian vector bundle

Article copyright:
© Copyright 1982
American Mathematical Society