1. Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions Jan Kříž QMath9, Giens13 September 2004

2. 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.

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 L 2 (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) L 2 (W), Dom Q := {y  W 1,2 (W) | y  D= 0 a.e.} D   W … Dirichlet b.c.

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

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

7. 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 ),  -[-L]-1  N  [-L]   >  : s disc  .         L  (0, L 0 ]  s disc = , L  L 0  s disc  .

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

11. L = 1/2

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

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

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

15. Configuration  :=   (0,d),  =((- ,-d)  {d})  ((d,  )  {d}), I:= (-d,d) N=(   {0})  (I  {d}) Operators -D W Q W (f,y) = (  f,  y ) L 2 (W),Dom Q W ={y  W 1,2 (W) | y  =0} Dom(-D W )... can be exactly determined -D I Q I (f,y) = (  f,  y ) L 2 (I),Dom Q I = W 0 1,2 (I) Dom(-D I ) ={y  W 2,2 (I) | y(-d) = y(d) = 0}

16. Energy spectrum 1. Nontrivial combination of b.c. in straight strips limit case of thin waveguides Discrete eigenvalues l i (d), i = 1,2,...,N d, where -[-L]-1  N d  - [-L]... eigenvalues of -D W m i, i  ... eigenvalues of -D I Theorem:  N  ,  e >0,  d 0 : (d < d 0 )  | l i (d) - m i | < e,  i = 1,..., N. PROOF: Kuchment, Zeng, J.Math. Anal.Appl. 258,(2001),671-700 Lemma1:R d : Dom Q I  Dom Q W, R d (f )(x,y) = f (x). f  Dom Q I :

17. Energy spectrum 1. Nontrivial combination of b.c. in straight strips limit case of thin waveguides Corollary 1:  i = 1,..., N, l i (d)  m i. PROOF: Min-max principle. W N (W)... linear span of N lowest eigenvalues of -D W. Lemma 2:T d : W N (W)  Dom Q I, T d (y )(x) = y (x,y)  I. for d small enough and y  W N (W): 1. 2. Corollary 2:  i = 1,..., N, m i  l i (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), 2574-2580. Goldstone, Jaffe, Phys.Rev.B 45 (1992), 14100-14107.

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

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

21. 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 width d  :  2  2 : {(s,u)  G(s) + u n(s)} W :=  (W o )...curved strip along G k  := max {0,  k} a :=   k(s) ds...bending angle

22. Curved strips - simplest combination of boundary conditions Assumptions: W is not self-intersecting k  L  (  ), d || k + ||  < 1.  : W o  W... C 1 – diffeomorphism  -1 defines natural coordinates (s,u). Hilbert space L 2 (W)  L 2 (W o, (1-u k(s)) ds du) Hamiltonian: unique s.a. operator H of 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 (W o ) | y(s,0) = 0 a.e. }

23. 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.

24. 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.

25. 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