Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions Jan Kříž QMath9, Giens 13 September 2004

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.

Model of quantum waveguide free particle of an effective mass living in nontrivial planar region Wof 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 -Dacting 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

ess 2d 2), ess 2d 2), Energy spectrum1. Nontrivial combination of b.c. in straight strips -[-L]-1 N [-L] -[-L]-1N[-L]     L (0 , L0]  sdisc= , L  L0 sdisc.   > : sdisc  .

Energy spectrum1. Nontrivial combination of b.c. in straight strips

Energy spectrum1. Nontrivial combination of b.c. in straight strips L = 1/2

Energy spectrum1. Nontrivial combination of b.c. in straight strips L = 2

Energy spectrum1. Nontrivial combination of b.c. in straight strips L=0.27

Energy spectrum1. Nontrivial combination of b.c. in straight stripslimit case of thin waveguides

Energy spectrum1. Nontrivial combination of b.c. in straight stripslimit case of thin waveguides • Configuration:=  (0,d), =((-,-d){d}) ((d, ) {d}) , I:= (-d,d)N=( {0}) (I{d}) • Operators • -DWQW(f,y) = (f, y )L2(W),Dom QW={yW1,2(W) | y =0} • Dom(-DW) ... can be exactly determined • -DIQI(f,y) = (f, y )L2(I),Dom QI = W01,2(I) Dom(-DI) ={y W2,2(I) | y(-d) = y(d) = 0}

Energy spectrum1. Nontrivial combination of b.c. in straight stripslimit case of thin waveguides • Discrete eigenvaluesli(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 spectrum1. Nontrivial combination of b.c. in straight stripslimit 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.

sess=  p24 d2) , ) sess= [ p2 / d2 , ) Energy spectrum2. Simplest combination of b.c. in curved strips sdisc , whenever the strip is curved. The existence of a discrete bound state essentially depends on the direction of the bending.

Energy spectrum2. Simplest combination of b.c. in curved strips sdisc sdisc,if d is small enough sdisc= 

Curved strips - simplest combination of boundary conditions • Configuration space G : 2...C2 - infinite plane curve n = (-G2’, G1’) ... 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 := (Wo)...curved strip along G k:= max {0,k} a := k(s) ds ... bending angle

Curved strips - simplest combination of boundary conditions • Assumptions:Wis not self-intersecting k L(), d ||k+|| < 1.  : Wo W ... C1 – diffeomorphism -1 defines natural coordinates (s,u). Hilbert space L2(W) L2(Wo, (1-u k(s)) ds du) • Hamiltonian: unique s.a. operator H of quadratic form ____ _____ Q(,f) := (Wo (1-u k(s))-1sy sf + (1-u k(s)) uy uf )ds du Dom Q := {y W1,2 (Wo) | y(s,0) = 0 a.e.}

Curved strips - simplest combination of boundary conditions • Essential spectrum: Theorem: lim|s| k(s) = 0 sess(H) = [p/(4d2), ). PROOF: 1. DN – bracketing 2. Generalized Weyl criterion (Deremjian,Durand,Iftimie, Commun. in Parital Differential Equations 23 (1998), no. 1&2, 141-169.

Curved strips - simplest combination of boundary conditions • Discrete spectrum: Theorem: (i) Assume k  0.If one of (a) k L1() and a  0, (b) k-  0 and d is small enough, is satisfied then inf s(H) < p/(4d2). (ii) If k-  0 then inf s(H)  p/(4d2). PROOF: (i) variationally (ii)  y  Dom Q : Q(y, y) - p/(4d2) ||y||2 0. Corollary: Assume lim|s| k(s) = 0. Then (i) Hhas an isolated eigenvalue. (ii) sdisc(H)is empty.

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

Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions