240 likes | 382 Views
Convergence of Spectra of quantum waveguides with combined boundary conditions. Jan K říž M 3 Q, Bressanone 21 February 2005. Collaboration with Jaroslav Dittrich and David K rejčiřík (NPI AS CR , Řež near Prague).
E N D
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 Wof 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 -Dacting in L2(W)(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 Q(y,f) := ( y,f)L2(W), Dom Q := {y W1,2(W) |yD= 0 a.e.} D W… Dirichlet b.c.
Energy spectrum 1. Nontrivial combination of b.c. in straight strips
Energy spectrum 1. Nontrivial combination of b.c. in straight strips L d /d
ess 2d 2), ess 2d 2), Energy spectrum1. Nontrivial combination of b.c. in straight strips -[-L]-1 N [-L] -[-L]-1N[-L] L (0 , L0] sdisc= , L L0 sdisc. > : sdisc .
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 L = 1/2
Energy spectrum1. Nontrivial combination of b.c. in straight strips L = 2
Energy spectrum1. Nontrivial combination of b.c. in straight strips L=0.27
Energy spectrum1. Nontrivial combination of b.c. in straight stripslimit case of thin waveguides
Energy spectrum1. Nontrivial combination of b.c. in straight stripslimit case of thin waveguides • Configuration:= (0,d), =((-,-d){d}) ((d, ) {d}) , I:= (-d,d)N=( {0}) (I{d}) • Operators • -DWQW(f,y) = (f, y )L2(W),Dom QW={yW1,2(W) | y =0} • Dom(-DW) ... can be exactly determined • -DIQI(f,y) = (f, y )L2(I),Dom QI = W01,2(I) Dom(-DI) ={y W2,2(I) | y(-d) = y(d) = 0}
Energy spectrum1. Nontrivial combination of b.c. in straight stripslimit case of thin waveguides • Discrete eigenvaluesli(d), i = 1,2,...,Nd, where -[-L]-1 Nd -[-L]...eigenvalues of -DW • mi , i ...eigenvalues of -DI • Theorem: N , e >0, d0 : (d < d0 ) |li(d) -mi| < e,i = 1, ..., N. • PROOF:Kuchment, Zeng, J.Math. Anal.Appl. 258,(2001),671-700 • Lemma1: Rd: Dom QI Dom QW, Rd(f )(x,y) = f (x). • f Dom QI :
Energy spectrum1. Nontrivial combination of b.c. in straight stripslimit case of thin waveguides • Corollary 1: i = 1, ..., N, li(d) mi . • PROOF: Min-max principle. • WN(W) ...linear span of N lowest eigenvalues of -DW . • Lemma 2: Td: WN(W) Dom QI , Td(y )(x) = y (x,y)I . for d small enough and y WN(W): 1. 2. • Corollary 2: i = 1, ..., N, mili(d) (1 + O(d))+ O(d).
Energy spectrum 2. 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.
sess= p24 d2) , ) sess= [ p2 / d2 , ) Energy spectrum2. Simplest combination of b.c. in curved strips sdisc , whenever the strip is curved. The existence of a discrete bound state essentially depends on the direction of the bending.
Energy spectrum2. Simplest combination of b.c. in curved strips sdisc sdisc,if d is small enough sdisc=
Energy spectrum2. Simplest combination of b.c. in curved strips:limit case of thin waveguides Dirichlet b.c. inf sess- inf s = - l(k)+ O(d), l(k)…1. eigenvalue of the operator -D -k2 / 4on L2(), k… curvature of the boundary curve Duclos, Exner, Rev.Math.Phys. 7(1995), 73-102. Combined b.c.(WG with k having bounded support) inf sess-inf s - a/(l d)+ O(d-1/2), a = k(s) ds…bending angle, l … length of the support of k.
Energy spectrum2. Simplest combination of b.c. in curved strips:limit case of mildly curved waveguides k = b k0, a = b a0. Dirichlet b.c. inf s =inf sess- Cb4+O(b5), Duclos, Exner, Rev.Math.Phys. 7(1995), 73-102. Combined b.c.(WG with k having bounded support) inf sinf sess- (3a2) / (8d3)b2+O(b3)
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