M3Q, Bressanone 21 February 2005 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.

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

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

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

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

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

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}

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),671-700 Lemma1: Rd: Dom QI  Dom QW, Rd(f )(x,y) = f (x). f  Dom QI :

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).

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 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.

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

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), 73-102. 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.

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), 73-102. Combined b.c. (WG with k having bounded support) inf s  inf sess - (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