1. M3Q, Bressanone 21 February 2005
Convergence of Spectra of quantum waveguides with combined boundary conditions
Jan Kříž
M3Q, Bressanone 21 February 2005

2. 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),
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

3. 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)

4. 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.

5. 1. Nontrivial combination of b.c. in straight strips
Energy spectrum
1. Nontrivial combination of b.c. in straight strips

6. Evans, Levitin, Vassiliev, J.Fluid.Mech. 261 (1994), 21-31.

7. 1. Nontrivial combination of b.c. in straight strips
Energy spectrum
1. Nontrivial combination of b.c. in straight strips
L  d /d

8. Energy spectrum 1. Nontrivial combination of b.c. in straight strips
ess  2d 2),
ess  2d 2),
-[-L]-1  N  [-L]
-[-L]-1  N  [-L]
      
L  (0 , L0]  sdisc = ,
L  L0  sdisc  .
  >  : sdisc  .

9. Energy spectrum 1. Nontrivial combination of b.c. in straight strips

10. Energy spectrum 1. Nontrivial combination of b.c. in straight strips

11. Energy spectrum 1. Nontrivial combination of b.c. in straight strips

12. Energy spectrum 1. Nontrivial combination of b.c. in straight strips

13. Energy spectrum 1. Nontrivial combination of b.c. in straight strips

14. Energy spectrum 1. Nontrivial combination of b. c
Energy spectrum 1. Nontrivial combination of b.c. in straight strips limit case of thin waveguides

15. Energy spectrum 1. Nontrivial combination of b. c
Energy spectrum 1. Nontrivial combination of b.c. in straight strips limit case of thin waveguides
Configuration  :=   (0,d), =((-,-d) {d})  ((d, ) {d}) , I:= (-d,d) N=(  {0})  (I {d})
Operators
-DW QW(f,y) = (f, y )L2(W) , Dom QW={yW1,2(W) | y =0}
Dom(-DW) ... can be exactly determined
-DI QI(f,y) = ( f, y )L2(I) , Dom QI = W01,2(I)
Dom(-DI) ={y  W2,2(I) | y(-d) = y(d) = 0}

16. Energy spectrum 1. Nontrivial combination of b. c
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   ... eigenvalues of -DI
Theorem:  N   ,  e >0,  d0 : (d < d0 )  | li(d) - mi| < e,  i = 1, ..., N.
PROOF: Kuchment, Zeng, J.Math. Anal.Appl. 258,(2001),
Lemma1: Rd: Dom QI  Dom QW, Rd(f )(x,y) = f (x).
f  Dom QI :

17. Energy spectrum 1. Nontrivial combination of b. c
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 , Td(y )(x) = y (x,y) I .
for d small enough and y  WN(W): 1. 2.
Corollary 2:  i = 1, ..., N, mi li(d) (1 + O(d)) + O(d).

18. Energy spectrum 2. Simplest combination of b.c. in curved strips
asymptotically straight strips
Exner, Šeba, J.Math.Phys. 30 (1989),
Goldstone, Jaffe, Phys.Rev.B 45 (1992),

19. Energy spectrum 2. Simplest combination of b.c. in curved strips
sess =  p2  4 d 2) , )
sess = [ p2 / d 2 , )
sdisc , whenever the strip is curved.
The existence of a discrete bound state essentially depends on the direction of the bending.

20. Energy spectrum 2. Simplest combination of b.c. in curved strips
sdisc  
sdisc  , if d is small enough
sdisc = 

21. Energy spectrum 2. Simplest combination of b. c
Energy spectrum 2. Simplest combination of b.c. in curved strips: limit case of thin waveguides
Dirichlet b.c.
inf sess - inf s = - l(k) + O(d),
l(k) … 1. eigenvalue of the operator -D -k2 / 4 on L2(),
k … curvature of the boundary curve
Duclos, Exner, Rev.Math.Phys. 7 (1995),
Combined b.c. (WG with k having bounded support)
inf sess - inf s  - a/(l d) + O(d-1/2),
a = k(s) ds … bending angle,
l … length of the support of k.

22. Energy spectrum 2. Simplest combination of b. c
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 s = inf sess - C b4 + O(b5),
Duclos, Exner, Rev.Math.Phys. 7 (1995),
Combined b.c. (WG with k having bounded support)
inf s  inf sess - (3a2) / (8d3) b2 +O(b3)

23. 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