Spectral Properties of Planar Quantum Waveguides with

Combined Boundary Conditions

Jan Kříž

Tokyo Metropolitan University, 19 January 2004

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.

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

Spectral Properties of

• What are the quantum waveguides?semiconductor (GaAs – AlGaAs) or metallic microstructures of the tube like shape

(a) small size 10 nm;(b) high purity (emean free path m);

(c) crystallic structure.

mesoscopic physics

free particle of an effective mass living in nontrivial planar region

Planar Quantum Waveguides with

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 withCombined boundary conditions

Spectral Properties of

• Mathematical point of view

spectrum of acting in the Hilbert space L2(putting physical constants equaled to 1)

Planar Quantum Waveguides withCombined boundary conditions

Hamiltonian

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

• Quadratic form

Q:=

Dom Q := { W1,2() = 0 a.e.}

Dirichlet b.c.

• Question: exact form of the operator domain;

Dom W2,2() satisfies b.c.

Examples of “ugly” regions

Dom W2,2() satisfies b.c.

fD(r,) = (r) r sin (),

C(r) = 1 … for r (r) = 0 … for r

fDN(r,) = (r) r sin ()

O.V.GusevaBirman,Skvortsov, Izv.Vyssh.Uchebn.Zaved.,Mat.30(1962),12-21.

Examples of “ugly” regions

Dom W2,2() satisfies b.c.

distance of centers of discs … at least 2radii of discs … 1/n for n = 1,2,3,…

fn(rnnn) (rnln n + ln rn

f = n=1 fn

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

d

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

ess d ess d

NN

disc

disc

disc

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 strips

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

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

Energy spectrum2. 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 spectrum2. Simplest combination of b.c. in curved strips

ess d ess d

The existence of a discrete bound state

essentially depends on the direction of the

bending.

disc whenever the strip is curved.

Energy spectrum2. Simplest combination of b.c. in curved strips

disc

disc if d is small enough

disc

Curved strips - simplest combination of boundary conditions

• Configuration space

2...C2infinite plane curve

2’, 1’’) ... unit normal vector field

det (’’’...curvature

d) ... straight strip of the length d

: 22 : {(s,u) (s) + u (s)}

...curved strip along

max {0,

(s) ds ... bending angle

Curved strips - simplest combination of boundary conditions

• Assumptions: is not self-intersecting

L(), d|| ||

: ... C1 – diffeomorphism

-1 defines natural coordinates (s,u).

Hilbert space L(L(, (1u (s)) ds du)

• Hamiltonian: unique s.a. operator H quadratic form

____ _____

Q() := (o (1u (s))-1 ss(1u (s)) uu)ds du

Dom Q := {W1,2 () | (s,0) = 0 a.e.}

Curved strips - simplest combination of boundary conditions

• Essential spectrum:

Theorem: lim|s|(s) = 0 ess(H) = [(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 If one of

(a) L() and (b) - and d is small enough,

is satisfied then inf (H) < (4d2).

(ii) If - then inf (H) (4d2).

PROOF: (i) variationally(ii) Dom Q : Q(4d2) ||||2

Corollary: Assume lim|s|(s) = 0. Then (i) Hhas an isolated eigenvalue.

(ii) discHis 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