The effective Hamiltonian in curved quantum
waveguides as a consequence of strong resolvent
convergence
Helena Šediváková
Faculty of Nuclear Sciences and Physical Engineering
Czech Technical University in Prague
Joint author: David Krejčiřı́k (NPI, Řež)
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC 1/15
Motivation: Constrained quantum systems
The behavior of the quantum system on a
submanifold depends on the properties
of the ambient Riemannian manifold,
on the extrinsic curvature of the
submanifold or the shape of constraining
potential - quantum effects
(consequence of uncertainty principle).
Constrain
t su b
man
ifol
d
3
IR - Ambient space
Application: Molecular dynamics, quantum waveguides etc.
[JK] H. Jensen and H. Koppe. Quantum mechanics with constraints, 1971.
[To] J. Tolar. On a quantum mechanical d’Alembert principle, 1988.
[FH] R. Froese and I. Herbst. Realizing holonomic constraints in classical and
quantum mechanics, 2001.
[Mi] K. A. Mitchell. Gauge fields and extrapotentials in constrained quantum
systems, 2001.
[WT] J. Wachsmuth and S. Teufel. Effective Hamiltonians for constrained
quantum systems, 2009.
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC 2/15
Motivation: Quantum waveguides
The model for the quantum systems where the electron moves freely in
microstructures of long thin tubular shapes.
The existence of curvature induced bound states proved in [EŠ].
[EŠ] P. Exner and P. Šeba. Bound states in curved quantum waveguides, 1989.
[DE] P. Duclos and P. Exner. Curvature-induced bound states in quantum waveguides
in two and three dimensions, 1995.
[Po] O. Post. Branched quantum wave guides with Dirichlet boundary conditions:
the decoupling case, 2005.
[KK] D. Krejčiřı́k and J. Křı́ž. On the spectrum
of curved planar waveguides, 2005.
[BMT] G. Bouchitté, M. L. Mascarenhas, and L.
Trabucho. On the curvature and torsion
effects in one dimensional waveguides, 2007.
[BC] D. Borisov, and G. Cardone. Complete
asymptotic expansions for eigenvalues of
Dirichlet Laplacian in thin three-dimensional
rods, 2010
Source: http://www.nano.physik.uni-muenchen.de/research/rep99/Bert/bert.html
What are the bound states caused by?
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC 3/15
Motivation: Quantum waveguides
N(s)
T(s)
3
3
u=-1
Г(s)
u=1
Ω3
ε
Hamiltonian of a free particle: H = −∆Ω
D
Curvilinear coordinates + Unitary transformation
−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
ε-dependent Hamiltonian acting on fixed domain Ω
ε→0
−−−→
Effective Hamiltonian:
H eff = 1 ⊗ −
κ2
1 (−1,1)
Γ
∆
+
(−∆
−
)⊗1
D
D
ε2
4
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC 4/15
Motivation
[DE] P. Duclos, P. Exner : Curvature-induced bound states in quantum
waveguides in two and three dimensions, 1995.
Method: Standard perturbation theory.
It is assumed
κ ∈ C 2.
[BMT] G. Bouchitté, M. L. Mascarenhas, L. Trabucho: On the curvature
and torsion effects in one dimensional waveguides, 2007.
Method: Γ-convergence.
No smoothness assumption on the curvature:
κ ∈ L∞ .
к=0
R
_1
к= R
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC 5/15
Strip in plane
N(s)
T(s)
3
3
u=-1
Г(s)
u=1
Ω3
The strip along the curve Γ:
Ωε := L(Ω)
L : Ω → R2 : {(s, u) 7−→ Γ(s) + εuN(s)},
We assume that L is injective.
Ω = I × (−1, 1), I finite, infinite or semi-infinite
The metric tensor of the strip:
Gij
=
(∂i L) · (∂j L) =
|G |
=
ε2 (1 − εuκ(s))2
(1 − εuκ(s))2
0
0
ε2
κ(s) := det(Γ̇, Γ̈), κ supposed to be uniformly continuous and bounded.
Why was the assumption κ ∈ C 2 needed in the papers like [DE]?
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC 6/15
The transformation of the Hamiltonian
Hamiltonian on the curved strip in cartesian coordinates x ∈ Ωε - the
Dirichlet Laplacian on L2 (Ωε , dx):
ε
H = −∆Ω
D .
Hamiltonian in coordinates (s, u) ∈ Ω = I × (−1, 1):
H̃ε = −|G |−1/2 ∂i |G |1/2 G ij ∂j ,
(Hilbert space: L2 (Ω, |G |1/2 dsdu))
“straightening” of the strip using the unitary transformation U:
Uψ = |G |1/4 ψ
Ĥε = U H̃ε U −1 = |G |1/4 H̃ε |G |−1/4 = ∂i G ij ∂j + V
(Hilbert space: L2 (Ω, dsdu)).
Recall |G | = ε2 (1 − εuκ)2 !
How to get rid of the derivatives of κ in the formulae for Ĥε ?
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC 7/15
Our strategy: 1) The function fε
R s+ε
fε (s) :=
s
κ(ξ)dξ
ε
κ(s)= |s|
fε(s) : ε=1
ε=0.5
ε=0.25
Close to κ in the limit of thin strip:
ε→0
sup |κ(s) − fε (s)| ≤ sup |κ(s + ε) − κ(s)| −−−→ 0.
s∈I
s∈I
κ(s + ε) − κ(s)
f˙ε (s) =
.
ε
We use this function in the unitary transformation:
Differentiable:
˜ = Ũ H̃ Ũ −1 = |G̃ |1/4 H̃ |G̃ |−1/4
H̃
ε
ε
ε
where
|G̃ | = ε2 (1 − εufε )2 .
1−εκu
The Hilbert space L2 (Ω, 1−εf
dsdu) =: Hε is dependent on ε!
εu
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC 8/15
Our strategy: 2) Working with quadratic forms
˜ ) = W 1,2 (Ω)
∀ψ(s, u) ∈ Dom (Q̃
ε
0
˜ [ψ]
Q̃
ε
Z
|∂s ψ|2
1
(1 − εuκ)
dsdu + 2
dsdu
|∂u ψ|2
(1
−
εuκ)(1
−
εuf
)
ε
(1
− εufε )
ε
Ω
Ω
Z
+
Vε1 + Vε2 |ψ|2 dsdu +
ZΩ
u (κ(s + ε) − κ(s))
Re ψ̄∂s ψ dsdu
+
2
Ω (1 − εufε ) (1 − εuκ)
Z
=
where
Vε1 (s, u)
:=
Vε2 (s, u)
:=
1
2 κfε
(1 − εufε
1 2
4u
)2
−
(1 − εufε
3 2
4 fε
3
) (1 −
εuκ)−1
2
(κ(s + ε) − κ(s))
.
(1 − εufε )3 (1 − εuκ)
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC 9/15
Renormalization
The asymptotic of the eigenvalues:
˜ ) = E1 + O(1).
λn (H̃
ε
ε2
(−1,1)
E1 - the first eigenvalue of the transverse Laplacian −∆D
.
To get the interesting part of spectra we renormalize
˜ − E1 .
Hε = H̃
ε
ε2
Hε can be in the limit of thin strip approximated by a Hamiltonian
acting on I :
κ2
H eff = −∆ID − .
4
In what sense does H eff approximate Hε ?
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC
10/15
The generalized strong resolvent convergence
We introduce
ψε := (Hε + k)−1 F
for any F ∈ Hε , the resolvent equation: (Hε + k)ψε = F .
We decompose
ψε = P 1 ψε + (1 − P 1 )ψε = ϕ ⊗ χ1 + ω.
(−1,1)
χ1 - the eigenvector corresponding with the eigenvalue E1 of −∆D
P
1
- the projection on subspace
We show that
⇒
We find
,
n
o
H1ε = ψ ∈ Hε | ∃ϕ ∈ L2 (I ), ψ(s, u) = ϕ(s)χ1 (u) .
s
ω
−−−→ 0
in W 1,2 (Ω),
ϕ
−−−→ ϕ0
in W 1,2 (I )
ε→0
w
ε→0
w
ψε −−−→ ϕ0 ⊗ χ1
ε→0
ϕ0 = H eff + k
in W 1,2 (Ω).
−1
F1
where F 1 = π −1 P 1 F .
π : L2 (I ) → Hε1 , ϕ 7→ ϕ ⊗ χ1 .
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC
11/15
The generalized strong resolvent convergence
Theorem
Let Hε be the Hamiltonian unitarily equivalent to the shifted Dirichlet
E1
2
ε
Laplacian −∆Ω
D − ε2 on the strip of width 2ε along the curve Γ : I → R and
eff
let H be the one dimensional effective Hamiltonian acting on I :
κ2
H eff = −∆ID −
4
(κ is the curvature of Γ). Assume in addition that
(i) the strip is non self-intersecting, and
(ii) the curvature κ is uniformly continuous and bounded.
Then
ε→0
χI 0 (Hε + k)−1 − π(H eff + k)−1 π −1 P 1 F −
−−→ 0
Hε
∀F ∈ Hε and for every bounded I 0 ⊆ I .
What does this result yield?
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC
12/15
The consequences of the strong resolvent convergence
It is known:
Let Tn and T be the self-adjoint operators acting on the fixed Hilbert space H
and
ε→0
k(Tn + k)−1 F − (T + k)−1 F kH −−−→ 0
∀F ∈ H and some k ∈ ρ(T ). Then:
I finite: We get the convergence of all the eigenvalues and eigenfunctions
in norm.
I infinite: According to [We] this yields the convergence of eigenvalues
bellow the bottom of essential spectrum and the convergence of the
corresponding eigenfunctions in norm.
However, the consequences of our generalized strong resolvent convergence
have to be investigated in more detail.
[We] J. Weidmann. Continuity of the eigenvalues of self-adjoint operators with respect to the
strong operator topology. Integral Equations and Operator Theory, 3/1:138-142, 1980.
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC
13/15
Conclusions
We have
proved the convergence of Hε to H eff with respect to the generalized strong
resolvent convergence under the condition
κ ∈ C 0 (Ī ),
established some spectral results of this convergence in case I finite.
The main idea of the proof was introducing
R s+ε
fε (s) :=
s
κ(ξ)dξ
.
ε
In future we would like to
find the consequences of the generalized strong resolvent convergence for case I
infinite,
prove the norm resolvent convergence,
extend these results on 3D waveguides.
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC
14/15
Thank you for your attention!
Helena Šediváková(FNSPE, Prague): The effective Hamiltonian in curved quantum waveguides as a consequence of SRC
15/15