Convergence of Spectra of quantum waveguides with combined boundary conditions

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 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)

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.

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 2 ess d 2

N N

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

limit case of thin waveguides

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

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

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

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

essd essd

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

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

limit case of thin waveguides

Dirichlet b.c.

inf ess inf d… 1. eigenvalue of the operator on L … curvature of the boundary curve

Duclos, Exner, Rev.Math.Phys. 7 (1995), 73-102.

Combined b.c. (WG with having bounded support)

inf essinf l dOd-1/2

sds… bending angle

l … length of the support of

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

limit case of mildly curved waveguides

Dirichlet b.c.

inf inf ess C ODuclos, Exner, Rev.Math.Phys. 7 (1995), 73-102.

Combined b.c. (WG with having bounded support)

inf inf ess8d3O

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