convergence of spectra of quantum waveguides with combined boundary conditions
DESCRIPTION
Convergence of Spectra of quantum waveguides with combined boundary conditions. Jan K říž M 3 Q, Bressanone21 February 2005. Collaboration with Jaroslav Dittrich and David K rejčiřík (NPI AS CR , Řež near Prague). - PowerPoint PPT PresentationTRANSCRIPT
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