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 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
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
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.}
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