1. Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions Jan Kříž Tokyo Metropolitan University, 19 January 2004

2. Joint work 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.

3. Program of the seminar Introduction: physical background Hamiltonian: definition, operator domain Summary of spectral results: comparison of our results with known ones Curved wires: precise statements and proofs Conclusions

4. Spectral Properties of What are the quantum waveguides? semiconductor (GaAs – AlGaAs) or metallic microstructures of the tube like shape (a) small size 10  100 nm ; (b) high purity( e  mean free path  m m );(c) crystallic structure. mesoscopic physics free particle of an effective mass living in nontrivial planar region W Planar Quantum Waveguides with

5. Spectral Properties of 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 Planar Quantum Waveguides with Combined boundary conditions

6. Spectral Properties of Mathematical point of view spectrum of  D acting in the Hilbert space L 2 (W) (putting physical constants equaled to 1) Planar Quantum Waveguides with Combined boundary conditions

7. 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 ) 2, Dom Q := {   W 1,2 (  )     = 0 a.e.} D  ... Dirichlet b.c. Question: exact form of the operator domain; Dom (  D)     W 2,2 (  )   satisfies b.c. 

8. Examples of “ugly” regions Dom (-D)     W 2,2 (  )   satisfies b.c.  f D (r,q) = x(r) r b sin (b q),  C  ((0,  )) x(r) = 1 … for r  (0,1/3) x(r) = 0 … for r  (2/3,  ) f DN (r,q) = x(r) r 1/2 sin (q/2) O.V.Guseva Birman,Skvortsov, Izv.Vyssh.Uchebn.Zaved.,Mat.30(1962),12-21.

9. Examples of “ugly” regions Dom (-D)     W 2,2 (  )   satisfies b.c.  distance of centers of discs … at least 2 radii of discs … 1/n for n = 1,2,3,… f n (r n,q n ) = -(1/n) x (r n ) (ln n + ln r n )   f =  n=1 f n

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

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

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

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

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

16. L = 1/2

17. Energy spectrum 1. Nontrivial combination of b.c. in straight strips L = 2

18. Energy spectrum 1. Nontrivial combination of b.c. in straight strips L=0.2 7

19. 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.

20. Energy spectrum 2. Simplest combination of b.c. in curved strips  ess    2   d 2 ),   ess    2  d 2,  The existence of a discrete bound state essentially depends on the direction of the bending.  disc  , whenever the strip is curved.

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

22. Curved strips - simplest combination of boundary conditions Configuration space G :  2...C 2 - infinite plane curve n = (-G 2 ’, G 1 ’’)...unit normal vector field k = det (G’,G’’)...curvature  o :=   (0,d)...straight strip of the length d  :  2  2 : {(s,u)  G(s) + u n(s)}  :=  (  o )...curved strip along G k  := max {0,  k} a :=   k(s) ds...bending angle

23. Curved strips - simplest combination of boundary conditions Assumptions:  is not self-intersecting k  L  (  ), d||| k + ||  < 1.  :  o  ... C 1 – diffeomorphism  -1 defines natural coordinates (s,u). Hilbert space L 2 (  )  L 2 (  o, (1-u k(s)) ds du) Hamiltonian: unique s.a. operator H  quadratic form ____ _____ Q( ,f) := (  W o (1-u k(s)) -1  s y  s f + (1-u k(s))  u y  u f ) ds du Dom Q := { y  W 1,2 (  o ) | y(s,0) = 0 a.e. }

24. Curved strips - simplest combination of boundary conditions Essential spectrum: Theorem:lim |s|  k(s) = 0  s ess (H) = [ p/(4d 2 ),  ). PROOF: 1. DN – bracketing 2. Generalized Weyl criterion  (Deremjian,Durand,Iftimie, Commun. in Parital Differential Equations 23 (1998), no. 1&2, 141-169.

25. Curved strips - simplest combination of boundary conditions Discrete spectrum: Theorem: (i) Assume k  0. If one of (a) k  L 1 (  ) and a  0, (b) k -  0 and d is small enough, is satisfied then inf s(H) < p/(4d 2 ). (ii) If k -  0 then inf s(H)  p/(4d 2 ). PROOF: (i) variationally (ii)  y  Dom Q : Q(y, y) - p/(4d 2 ) ||y|| 2  0.  Corollary: Assume lim |s|  k(s) = 0. Then (i)  H has an isolated eigenvalue. (ii)  s disc ( H) is empty.

26. 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