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

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 říž QMath9, Giens 1 3 September 200 4. Collaboration with Jaroslav Dittrich (NPI AS CR , Řež near Prague) and David K rejčiřík (Instituto Superior Tecnico, Lisbon). - PowerPoint PPT Presentation

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Page 1: 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

Page 2: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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.

Page 3: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

Model of quantum waveguide

free particle of an effective mass living in nontrivial planar region 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 -acting in L2(putting physical constants equaled to 1)

Page 4: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

Hamiltonian

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

• Quadratic form

QL

Dom Q := {W a.e.}

… Dirichlet b.c.

Page 5: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

Energy spectrum

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

Page 6: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

Page 7: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

Energy spectrum

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

d

Page 8: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

ess d 2 ess d 2

N N

disc

disc

disc

Page 9: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

Page 10: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

Page 11: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

Page 12: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

Page 13: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

Page 14: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

limit case of thin waveguides

Page 15: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

limit case of thin waveguides

• Configuration d), d d , I d

• Operators

Q)L2(Dom QW1,2

Dom ... can be exactly determined

Q L2(IDom QW01,2

Dom) W2,2

Page 16: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

limit case of thin waveguides

• Discrete eigenvaluesi(d), i = 1,2,...,Nd, where Nd

eigenvalues of

i , i eigenvalues of I

Theorem: N d0 : (d < d0 ) i(d) i| i = 1, ..., N.

PROOF: Kuchment, Zeng, J.Math. Anal.Appl. 258,(2001),671-700

Lemma1: Rd: Dom QDom QRdx,yx

Dom Q 2

)(

2

)(

2

)(

2

)(

2

2

2

2

)(

)(

L

d

L

d

IL

IL

R

R

Page 17: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

limit case of thin waveguides

Corollary 1: i = 1, ..., N, i(d) i .

PROOF: Min-max principle.

WN(linear span of N lowest eigenvalues of

Lemma 2: Td: WN(Dom QTdxx,y

for d small enough and WN(

Corollary 2: i = 1, ..., N, i(d) (1 + O(d)) + O(d).

2

)(

2

)(

12

)(222

)()(

LLIL

d dOdT

)(12

)(

12

)(22

dOdTLIL

d

Page 18: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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.

Page 19: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

essd essd

The existence of a discrete bound state

essentially depends on the direction of the

bending.

disc whenever the strip is curved.

Page 20: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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

disc

disc if d is small enough

disc

Page 21: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

Curved strips - simplest combination of boundary conditions

• Configuration space...C2infinite plane curve’,’) ... unit normal vector fielddet’’’...curvatureod ... straight strip of the width d {(s,u) (s) + u (s)}o...curved strip along

max {0,(s) ds ... bending angle

Page 22: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

Curved strips - simplest combination of boundary conditions

• Assumptions: is not self-intersecting

Ld

o ... C1 – diffeomorphism

-1 defines natural coordinates (s,u).

Hilbert space LLou (s)) ds du)

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

____ _____

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

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

Page 23: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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.

Page 24: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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.

Page 25: Spectral Properties of Planar Quantum Waveguides with Combined Boundary Conditions

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