Convergence of Spectra of
quantum waveguides with
combined boundary conditions
Jan Kříž
M3Q, Bressanone
21 February 2005
Collaboration with Jaroslav Dittrich and
David Krejčiřík (NPI AS CR, Řež near
Prague)
• 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 = 
Energy spectrum
2. Simplest combination of b.c. in curved strips:
limit case of thin waveguides
Dirichlet b.c.
inf ess - inf  = - l(k) + O(d),
l(k) … 1. eigenvalue of the operator -D -k2
 on L2(),
k
… curvature of the boundary curve
Duclos, Exner, Rev.Math.Phys. 7 (1995), 73-102.
/
Combined b.c. (WG with k having bounded support)
inf ess - inf   - a/(l d) + O(d-1/2),
a
l
= k(s) ds
… bending angle,
… length of the support of k.
Energy spectrum
2. Simplest combination of b.c. in curved strips:
limit case of mildly curved waveguides
k
= b
k0,
a
=
b a0.
Dirichlet b.c.
inf  = inf ess - C b + O(b5),
Duclos, Exner, Rev.Math.Phys. 7 (1995), 73-102.
Combined b.c. (WG with k having bounded support)
inf   inf ess - (3a2) / (8d3) b2
+O(b3)
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