Fredholmness vs. Spectral Discreteness for first-order differential operators

Anghel, N.

doi:10.1090/proc12741

Fredholmness vs. Spectral Discreteness for first-order differential operators
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Proc. Amer. Math. Soc. 144 (2016), 693-701 Request permission

Abstract:

It is shown that for essentially self-adjoint first-order differential operators $D$, acting on sections of bundles over complete (non-compact) manifolds, Fredholmness vs.$\!$ Spectral Discreteness is the same as ‘$\exists c>0$, $D$ is $c$-invertible at infinity’ vs. ‘$\forall c>0$, $D$ is $c$-invertible at infinity’. An application involving the spectral theory of electromagnetic Dirac operators is then given.

References

Nicolae Anghel, An abstract index theorem on noncompact Riemannian manifolds, Houston J. Math. 19 (1993), no. 2, 223–237. MR 1225459
Paul R. Chernoff, Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Analysis 12 (1973), 401–414. MR 0369890, DOI 10.1016/0022-1236(73)90003-7
M. Gromov and B. Lawson, Positive Scalar Curvature and the Dirac Operator on Complete Riemannian Manifolds, Publ. Math. IHES, $\mathbf {58}$, 295-408, (1983).
Akira Iwatsuka, Magnetic Schrödinger operators with compact resolvent, J. Math. Kyoto Univ. 26 (1986), no. 3, 357–374. MR 857223, DOI 10.1215/kjm/1250520872
A. M. Molčanov, On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order, Trudy Moskov. Mat. Obšč. 2 (1953), 169–199 (Russian). MR 0057422
Naohiro Suzuki, Discrete spectrum of electromagnetic Dirac operators, Proc. Amer. Math. Soc. 128 (2000), no. 3, 819–825. MR 1628440, DOI 10.1090/S0002-9939-99-05073-X
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Oxford, at the Clarendon Press, 1946 (German). MR 0019765
Hermann Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), no. 2, 220–269 (German). MR 1511560, DOI 10.1007/BF01474161