We consider the Dirichlet Laplacian in tubular neighbourhoods of complete non-compact Riemannian manifolds immersed in the Euclidean space. We show that the essential spectrum coincides with the spectrum of a planar tube provided that the second fundamental form of the manifold vanishes at infinity and the transport of the cross-section along the manifold is asymptotically parallel. For low dimensions and codimension, the result applies to the location of propagating states in nanostructures under physically natural conditions.
REFERENCES
1.
N.
Charalambous
and Z.
Lu
, “On the spectrum of the Laplacian
,” Math. Ann.
359
(1–2
), 211
–238
(2014
).2.
G.
Carron
, P.
Exner
, and D.
Krejčiřík
, “Topologically nontrivial quantum layers
,” J. Math. Phys.
45
(2
), 774
–784
(2004
).3.
J.
Cheeger
and D. G.
Ebin
, Comparison Theorems in Riemannian Geometry
, North-Holland Mathematical Library
Vol. 9
(North-Holland Publishing Co.
, Amsterdam
, 1975
).4.
Z. H.
Chen
and Z. Q.
Lu
, “Essential spectrum of complete Riemannian manifolds
,” Sci. China Ser. A
35
(3
), 276
–282
(1992
).5.
B.
Chenaud
, P.
Duclos
, P.
Freitas
, and D.
Krejčiřík
, “Geometrically induced discrete spectrum in curved tubes
,” Differential Geom. Appl.
23
(2
), 95
–105
(2005
).6.
H. L.
Cycon
, R. G.
Froese
, W.
Kirsch
, and B.
Simon
, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry
, Springer Study Edition, Texts and Monographs in Physics
(Springer-Verlag
, Berlin
, 1987
).7.
Y.
Dermenjian
, M.
Durand
, and V.
Iftimie
, “Spectral analysis of an acoustic multistratified perturbed cylinder
,” Commun. Part. Differ. Equ.
23
(1–2
), 141
–169
(1998
).8.
H.
Donnelly
, “Exhaustion functions and the spectrum of Riemannian manifolds
,” Indiana Univ. Math. J.
46
(2
), 505
–528
(1997
).9.
P.
Duclos
, P.
Exner
, and D.
Krejčiřík
, “Bound states in curved quantum layers
,” Commun. Math. Phys.
223
(1
), 13
–28
(2001
).10.
J. F.
Escobar
, “On the spectrum of the Laplacian on complete Riemannian manifolds
,” Commun. Part. Differ. Equ.
11
(1
), 63
–85
(1986
).11.
J. F.
Escobar
and A.
Freire
, “The spectrum of the Laplacian of manifolds of positive curvature
,” Duke Math. J.
65
(1
), 1
–21
(1992
).12.
P.
Exner
and D.
Krejčiřík
, “Bound states in mildly curved layers
,” J. Phys. A
34
(30
), 5969
–5985
(2001
).13.
A.
Gray
, Tubes
, 2nd ed., Progress in Mathematics
Vol. 221
(Birkhäuser Verlag
, Basel
, 2004
) (With a preface by Vicente Miquel).14.
V.
Iftimie
, private communication (2002
).15.
D.
Krejčiřík
, Guides d'ondes Quantiques Bidimensionnels
, Facultas Mathematica Physicaque, Universitas Carolina Pragensis; Faculté des Sciences et Techniques
, Université de Toulon et du Var
, 2001
, Supervisors P. Duclos and P. Exner.16.
D.
Krejčiřík
, “Twisting versus bending in quantum waveguides
,” Analysis on Graphs and its Applications
, Proceedings of Symposia in Pure Mathematics
Vol. 77
(American Mathematical Society
, Providence, RI
, 2008
), pp. 617
–637
; e-print arXiv:0712.3371v2 [math-ph].17.
D.
Krejčiřík
and J.
Kříž
, “On the spectrum of curved quantum waveguides
,” Publ. RIMS, Kyoto University
41
, 757
–791
(2005
).18.
W.
Kühnel
, Differential Geometry: Curves—Surfaces—Manifolds
, Student Mathematical Library Vol. 16, Translated from the 1999 German original by Bruce Hunt
(American Mathematical Society
, Providence, RI
, 2002
).19.
J. Y.
Li
, “Spectrum of the Laplacian on a complete Riemannian manifold with nonnegative Ricci curvature which possess a pole
,” J. Math. Soc. Jpn.
46
(2
), 213
–216
(1994
).20.
C.
Lin
and Z.
Lu
, “On the discrete spectrum of generalized quantum tubes
,” Commun. Part. Differ. Equ.
31
(10-12
), 1529
–1546
(2006
).21.
C.
Lin
and Z.
Lu
, “Existence of bound states for layers built over hypersurfaces in
,” $\mathbb {R}^{n+1}$
J. Funct. Anal.
244
(1
), 1
–25
(2007
).22.
J. T.
Londergan
, J. P.
Carini
, and D. P.
Murdock
, Binding and Scattering in Two-Dimensional Systems
, Lecture Notes in Physics Monographs
Vol. 60
(Springer
, Berlin
, 1999
).23.
Z.
Lu
and D.
Zhou
, “On the essential spectrum of complete non-compact manifolds
,” J. Funct. Anal.
260
(11
), 3283
–3298
(2011
).24.
J.
Milnor
, Morse Theory (Based on lecture notes by M. Spivak and R. Wells)
, Annals of Mathematics Studies
Vol. 51
(Princeton University Press
, Princeton, NJ
, 1963
).25.
J. W.
Milnor
, “On the total curvature of knots
,” Ann. of Math.
52
(2
), 248
–257
(1950
).26.
M.
Reed
and B.
Simon
, Methods of Modern Mathematical Physics, I. Functional Analysis
(Academic Press
, New York
, 1972
).27.
J.
Wachsmuth
and S.
Teufel
, “Effective Hamiltonians for constrained quantum systems
,” Memoirs of the AMS
230
(1083
) (2013
).28.
J.
Wang
, “The spectrum of the Laplacian on a manifold of nonnegative Ricci curvature
,” Math. Res. Lett.
4
(4
), 473
–479
(1997
).29.
J.
Weidmann
, Linear Operators in Hilbert Spaces
, Graduate Texts in Mathematics Vol. 68, translated from the German by Joseph Szücs
(Springer-Verlag
, New York
, 1980
).30.
D. T.
Zhou
, “Essential spectrum of the Laplacian on manifolds of nonnegative curvature
,” Internat. Math. Res. Notices
1994
, 209
–214
(1994
).© 2014 AIP Publishing LLC.
2014
AIP Publishing LLC
You do not currently have access to this content.
