Let Ω be an unbounded two dimensional strip on a ruled surface in , n > 1. Consider the Laplacian operator in Ω with Dirichlet and Neumann boundary conditions on opposite sides of Ω. We prove some results on the existence and absence of the discrete spectrum of the operator; which are influenced by the twisted and bent effects of Ω. Provided that Ω is thin enough, we show an asymptotic behavior of the eigenvalues. The interest in those considerations lies on the difference from the purely Dirichlet case. Finally, we perform an appropriate dilatation in Ω and we compare the results.
REFERENCES
1.
D.
Borisov
, P.
Exner
, R.
Gadyl’shin
, and D.
Kreejčiřík
, “Bound states in weakly deformed strips and layers
,” Ann. Henri Poincaré
2
(3
), 553
–572
(2001
).2.
D.
Borisov
, P.
Exner
, and R.
Gadyl’shin
, “Geometric coupling thresholds in a two-dimensional strip
,” J. Math. Phys.
43
, 6265
–6278
(2002
).3.
W.
Bulla
, F.
Gesztesy
, W.
Renger
, and B.
Simon
, “Weakly coupled bound states in quantum waveguides
,” Proc. Am. Math. Soc.
125
(5
), 1487
–1495
(1997
).4.
B.
Chenaud
, P.
Duclos
, P.
Freitas
, and D.
Krejčiřík
, “Geometrically induced discrete spectrum in curved tubes
,” Differ. Geom. Appl.
23
(2
), 95
–105
(2005
).5.
J.
Dittrich
and J.
Kříž
, “Bound states in straight quantum waveguides with combined boundary condition
,” J. Math. Phys.
43
(8
), 3892
–3915
(2002
).6.
J.
Dittrich
and J.
Kríz
, “Curved planar quantum wires with Dirichlet and Neumann boundary conditions
,” J. Phys. A: Math. Gen.
35
(20
), L269
–L275
(2002
).7.
P.
Duclos
and P.
Exner
, “Curvature-induced bound states in quantum waveguides in two and three dimensions
,” Rev. Math. Phys.
07
(01
), 73
–102
(1995
).8.
P.
Duclos
, P.
Exner
, and D.
Krejčiřík
, “Bound states in curved quantum layers
,” Commun. Math. Phys.
223
, 13
–28
(2001
).9.
P.
Exner
and P.
Seba
, “Bound states in curved quantum waveguides
,” J. Math. Phys.
30
(11
), 2574
–2580
(1989
).10.
P.
Exner
, P.
Šeba
, M.
Tater
, and D.
Vaněk
, “Bound states and scattering in quantum waveguides coupled laterally through a boundary window
,” J. Math. Phys.
37
(10
), 4867
–4887
(1996
).11.
L.
Friedlander
and M.
Solomyak
, “On the spectrum of the Dirichlet Laplacian in a narrow infinite strip
,” Am. Math. Soc. Transl.
225
, 103
–116
(2008
).12.
L.
Friedlander
and M.
Solomyak
, “On the spectrum of the Dirichlet Laplacian in a narrow strip
,” Isr. J. Math.
170
, 337
–354
(2009
).13.
J.
Goldstone
and R. L.
Jaffe
, “Bound states in twisting tubes
,” Phys. Rev. B
45
(24
), 14100
–14107
(1992
).14.
D.
Krejčiřík
, “Quantum strips on surfaces
,” J. Geom. Phys.
45
(1-2
), 203
–217
(2003
).15.
D.
Krejčiřík
, “Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions
,” ESAIM: Control, Optim. Calculus Var.
15
(3
), 555
–568
(2009
).16.
D.
Krejčiřík
and R. T.
de Aldecoa
, “Ruled strips with asymptotically diverging twisting
,” Ann. Henri Poincaré
19
(7
), 2069
–2086
(2018
).17.
D.
Krejčiřík
and J.
Kříž
, “On the spectrum of curved planar waveguides
,” Publ. Res. Inst. Math. Sci.
41
(3
), 757
–791
(2005
).18.
D.
Krejčiřík
and Z.
Lu
, “Location of the essential spectrum in curved quantum layers
,” J. Math. Phys.
55
(8
), 083520
(2014
).19.
D.
Krejčiřík
and K.
Zahradová
, “Quantum strips in higher dimensions
,” Oper. Matrices
14
(3
), 635
–665
(2020
).20.
W.
Renger
and W.
Bulla
, “Existence of bound states in quantum waveguides under weak conditions
,” Lett. Math. Phys.
35
(1
), 1
–12
(1995
).21.
H.
Kovařík
and D.
Krejčiřík
, “A Hardy inequality in a twisted Dirichlet-Neumann waveguide
,” Math. Nachr.
281
(8
), 1159
–1168
(2008
).22.
D.
Krejčiřík
, N.
Raymond
, J.
Royer
, and P.
Siegl
, “Reduction of dimension as a consequence of norm-resolvent convergence and applications
,” Mathematika
64
(2
), 406
–429
(2018
).23.
D.
Krejčiřík
, “Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions
,” Math. Bohem.
139
(2
), 185
–193
(2014
).24.
T.
Jecko
, “On the mathematical treatment of the Born-Oppenheimer approximation
,” J. Math. Phys.
55
(5
), 053504
(2014
).25.
J.
Lampart
and S.
Teufel
, “The adiabatic limit of the Laplacian on thin fibre bundles
,” in Microlocal Methods in Mathematical Physics and Global Analysis
, Trends in Mathematics
(Birkhäuser Springer
, Basel
, 2013
), pp. 33
–36
.26.
N.
Raymond
, Bound States of the Magnetic Schrödinger Operator
, EMS Tracts in Mathematics Vol. 27
(EMS Press
, 2017
).27.
J.
Wachsmuth
and S.
Teufel
, Effective Hamiltonians for Constrained Quantum Systems
, Memoirs of the American Mathematical Society
(American Mathematical Society
, 2014
), Vol. 230
, pp. 1083
.28.
T.
Kato
, “On the convergence of the perturbation method
,” J. Fac. Sci., Univ. Tokyo Sect. 1 Math., Astron., Phys., Chem.
6
(3
), 145
–226
(1951
).29.
R.
Bedoya
, C. R.
de Oliveira
, and A. A.
Verri
, “Complex Γ-convergence and magnetic Dirichlet Laplacian in bounded thin tubes
,” J. Spectr. Theory
4
(3
), 621
–642
(2014
).30.
P.
Freitas
and D.
Krejčiřík
, “Instability results for the damped wave equation in unbounded domains
,” J. Differ. Equations
211
(1
), 168
–186
(2005
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
