Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions. Jan K QMath9, Giens 1 3 September 200 4. Collaboration with Jaroslav Dittrich (NPI AS CR , e near Prague) and David K rejik (Instituto Superior Tecnico, Lisbon). - PowerPoint PPT Presentation

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  • Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

    Jan K

    QMath9, Giens13 September 2004

  • Collaboration with Jaroslav Dittrich (NPI AS CR, e near Prague) and David Krejik (Instituto Superior Tecnico, Lisbon)J. Dittrich, J. K, Bound states in straight quantum waveguides with combined boundary conditions, J.Math.Phys. 43 (2002), 3892-3915.J. Dittrich, J. K, Curved planar quantum wires with Dirichlet and Neumann boundary conditions, J.Phys.A: Math.Gen. 35 (2002), L269-L275.D. Krejik, J. K, On the spectrum of curved quantum waveguides, submitted, available on mp_arc, number 03-265.

  • Model of quantum waveguidefree particle of an effective mass living in nontrivial planar region W of the tube-like shape

    Impenetrable walls: suitable boundary conditionDirichlet 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 viewspectrum of -D acting in L2(W) (putting physical constants equaled to 1)

  • HamiltonianDefinition:one-to-one correspondence between the closed, symmetric, semibounded quadratic forms and semibounded self-adjoint operators

    Quadratic formQ(y,f) := ( y,f)L2(W), Dom Q := {y W1,2(W) | yD= 0 a.e.}D W Dirichlet b.c.

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

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

  • Energy spectrum1. Nontrivial combination of b.c. in straight stripsL d /d

  • Energy spectrum1. Nontrivial combination of b.c. in straight stripsess 2d 2),ess 2d 2),-[-L]-1 N [-L]-[-L]-1 N [-L] > : sdisc . L (0 , L0] sdisc = , L L0 sdisc .

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

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

  • Energy spectrum1. Nontrivial combination of b.c. in straight stripsL = 1/2

  • Energy spectrum1. Nontrivial combination of b.c. in straight stripsL = 2

  • Energy spectrum1. Nontrivial combination of b.c. in straight stripsL=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={yW1,2(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 waveguidesCorollary 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 spectrum2. Simplest combination of b.c. in curved stripsasymptotically straight stripsExner, eba, J.Math.Phys. 30 (1989), 2574-2580.Goldstone, Jaffe, Phys.Rev.B 45 (1992), 14100-14107.

  • Energy spectrum2. Simplest combination of b.c. in curved stripssess = p2 4 d 2) , )sess = [ p2 / d 2 , )The existence of a discrete bound state essentially depends on the direction of the bending.sdisc , whenever the strip is curved.

  • Energy spectrum2. Simplest combination of b.c. in curved stripssdisc sdisc , if d is small enoughsdisc =

  • Curved strips - simplest combination of boundary conditionsConfiguration spaceG : 2...C2 - infinite plane curven = (-G2, G1)...unit normal vector fieldk = det (G,G)...curvatureo := (0,d)...straight strip of the width d : 22 : {(s,u) G(s) + u n(s)}W := (Wo)...curved strip along Gk := max {0,k}a := k(s) ds...bending angle

  • Curved strips - simplest combination of boundary conditionsAssumptions:W is not self-intersectingk 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 conditionsEssential spectrum:Theorem:lim|s| k(s) = 0 sess(H) = [p/(4d2), ).PROOF: 1. DN bracketing2. Generalized Weyl criterion (Deremjian,Durand,Iftimie, Commun. in Parital Differential Equations 23 (1998), no. 1&2, 141-169.

  • Curved strips - simplest combination of boundary conditionsDiscrete 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) H has an isolated eigenvalue.(ii) sdisc(H) is empty.

  • ConclusionsComparison with known resultsDirichlet b.c. bound state for curved stripsNeumann b.c. discrete spectrum is emptyCombined b.c. existence of bound states depends on combination of b.c. and curvature of a stripOpen problemsmore complicated combinations of b.c.higher dimensionsmore general b.c. nature of the essential spectrum