Spectral Properties of Planar
Quantum Waveguides with
Combined Boundary Conditions
Jan Kříž
QMath9, Giens
13 September 2004
Collaboration with Jaroslav Dittrich (NPI AS
CR, Řež near Prague) and David Krejčiřík
(Instituto Superior Tecnico, Lisbon)
• J. Dittrich, J. Kříž, Bound states in straight quantum
waveguides with combined boundary conditions,
J.Math.Phys. 43 (2002), 3892-3915.
• J. Dittrich, J. Kříž, Curved planar quantum wires with
Dirichlet and Neumann boundary conditions,
J.Phys.A: Math.Gen. 35 (2002), L269-L275.
• D. Krejčiřík, J. Kříž, On the spectrum of curved
quantum waveguides, submitted, available on
mp_arc, number 03-265.
Model of quantum waveguide
free particle of an effective mass living in nontrivial
planar region W of the tube-like shape
Impenetrable walls: suitable boundary condition
• Dirichlet b.c. (semiconductor structures)
• Neumann b.c. (metallic structures, acoustic or
electromagnetic waveguides)
• Waveguides with combined Dirichlet and Neumann
b.c. on different parts of boundary
Mathematical point of view
spectrum of -D acting in L2(W) (putting physical
constants equaled to 1)
Hamiltonian
• Definition: one-to-one correspondence between the
closed, symmetric, semibounded quadratic forms and
semibounded self-adjoint operators
• Quadratic form
Q(y,f)
:= ( y,f)L2(W),
Dom Q := {y  W1,2(W)
| yD=
0 a.e.}
D  W … Dirichlet b.c.
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
Evans, Levitin, Vassiliev, J.Fluid.Mech. 261 (1994), 21-31.
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
L
= d
/d
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
ess =
2/(d
2),)
--L]-1  N 
-L]

L0  (0 ,
1) :
L  (0 ,
L0]  disc =
,
ess =
2/(d
2),)
--L]-1  N 
-L]

L
> 0 : disc
 .
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
L
= 1/2
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
L
=
2
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
L=0.2
7
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
limit case of thin waveguides
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
limit case of thin waveguides
• Configuration
W :=   (0,d),
D=((-,-d)
{d})  ((d, ) {d}) ,
I:=
(-d,d)
N=(  {0})  (I
{d})
• Operators
-DW QW(f,y) = (f, y )L2(W) ,
QW={yW1,2(W) | y D=0}
Dom(-DW) ...
Dom
can be exactly determined
-DI QI(f,y) =
QI =
W01,2(I)
Dom(-DI) ={y
( f,
y )L2(I) ,
Dom
 W2,2(I) | y(-d) =
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
limit case of thin waveguides
• Discrete eigenvalues
li(d), i = 1,2,...,Nd, where --L]-1  Nd  -L]
... eigenvalues of -DW
mi , i 
N ... eigenvalues of -DI
Theorem:  N  N ,  e >0,  d0 : (d <
d0 )  | li(d) - mi| < e,
 i = 1, ..., N.
PROOF: Kuchment, Zeng, J.Math. Anal.Appl. 258,(2001),671-700
Lemma1:
Rd: Dom QI  Dom QW,
Rd(f
2
2
d
= f (x).
R ( )
 L2 ( I )

2
f  Dom QI :

L2 ( I )
L2 (  )
R ( )
d
2
L2 (  )
)(x,y)
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
limit case of thin waveguides
Corollary 1:  i = 1, ..., N,
li(d)  mi .
PROOF: Min-max principle.
WN(W) ...
.
linear span of N lowest eigenvalues of -DW
Lemma 2: Td: WN(W)  Dom QI ,
)(x) = y (x,y)2 I .


for d small enough and y 
(T  )
d
1. T
2.

d
2
L (I )
2
L2 ( I )
d
2
1
L2 (  )
d 
1
2
L2 (  )
Td(y
 O( d ) 
WN(W):
1  O(d )
2
L2 (  )

Energy spectrum
2. Simplest combination of b.c. in curved strips
asymptotically straight strips
Exner, Šeba, J.Math.Phys. 30 (1989), 2574-2580.
Goldstone, Jaffe, Phys.Rev.B 45 (1992), 14100-14107.
Energy spectrum
2. Simplest combination of b.c. in curved strips
ess =  2 /
( d 2) ,
)
The existence of a
discrete bound state
essentially depends on
the direction of the
bending.
ess =
d
 2 /
2
, )
disc , whenever
the strip is curved.
Energy spectrum
2. Simplest combination of b.c. in curved strips
disc  
disc  , if d is small
enough
disc = 
Curved strips - simplest combination of
boundary conditions
• Configuration space
G : 2
...
curve
n = (-G2’, G1’) ...
field
k = det (G’,G’’)
Wo :=   (0,d)
the width d
L : 22 : {(s,u) 
W := L(Wo)
along G
k := max {0,k}
a :=  k(s) ds
C2
- infinite plane
unit normal vector
...
...
curvature
straight strip of
G(s) + u n(s)}
...
curved strip
...
bending angle
Curved strips - simplest combination of
boundary conditions
• Assumptions:
W is not self-intersecting
k  L(), d ||
< 1.
k+||
L : Wo  W
... C1 – diffeomorphism
L-1 defines natural coordinates (s,u).
Hilbert space L2(W)  L2(Wo, (1-u k(s)) ds
du)
• Hamiltonian: unique s.a. operator H of quadratic form
____
Q(y,f) := (Wo (1-u k(s))-1 sy
(1-u k(s)) uy uf )ds du
_____

sf
+
Curved strips - simplest combination of
boundary conditions
• Essential spectrum:
Theorem:
lim|s|
k(s) = 0 ess(H) = [/(4d2),
).
1. DN – bracketing
2. Generalized Weyl criterion
(Deremjian,Durand,Iftimie, Commun. in Parital Differential
Equations 23 (1998), no. 1&2, 141-169.
PROOF:
Curved strips - simplest combination of
boundary conditions
• Discrete spectrum:
Theorem: (i) Assume k  0. If one of
(a) k L1() and a  0,
(b) k-  0 and d is small enough,
is satisfied then inf (H) < /(4d2).
(ii) If k-  0 then inf (H)  /(4d2).
PROOF:
(i) variationally
(ii)  y  Dom Q : Q(y,
/(4d2) ||y||2 0.
Corollary: Assume lim|s| k(s) = 0. Then
(i) 
H
has an isolated eigenvalue.
(ii) 
disc(H) is empty.
y) -
Conclusions
• Comparison with known results
– Dirichlet b.c. bound state for curved strips
– Neumann b.c. discrete spectrum is empty
– Combined b.c. existence of bound states depends
on combination of b.c. and
curvature of a strip
• Open problems
– more complicated combinations of b.c.
– higher dimensions
– more general b.c.
– nature of the essential spectrum