Asymptotic behaviour of the heat equation in twisted waveguides

Transcription

1 Asymptotic behaviour of the heat equation in twisted waveguides Gabriela Malenová Faculty of Nuclear Sciences and Physical Engineering, CTU, Prague Nuclear Physics Institute, AS ČR, Řež Graphs and Spectra, Chemnitz, July 19, 2011 Jointly with: David Krejčiřík and Miloš Tater, NPI, AS ČR, Řež.

2 Layout Heat equation in twisted waveguide Quantum waveguides Hardy inequalities Heat equation on Ω θ Asymptotic behaviour Decay rate Self-similarity transformation Numerical solution Numerical methods Comparison The twisted case Conclusion

3 Introduction Straight waveguide Ω 0 := R ω, ω R 2 is non-circular cross-section Twisted waveguide Ω θ, ω is rotating with respect to non-constant angle θ(x 1 ). The Hamiltonian of a particle moving inside is descibed by the Dirichlet Laplacian Ω θ D : L2 (Ω θ ) L 2 (Ω θ ). It is associated with the quadratic form ψ ψ 2 with the domain D(Ω θ ) := H0 1(Ω θ).

4 Layout Heat equation in twisted waveguide Quantum waveguides Hardy inequalities Heat equation on Ω θ Asymptotic behaviour Decay rate Self-similarity transformation Numerical solution Numerical methods Comparison The twisted case Conclusion

5 Ω θ D and Ω 0 D Spectral stability have the same spectrum as a set: σ( Ω θ D ) = σ ess( Ω θ D ) = [E 1, ). E 1 is the threshold energy of ω D Difference: existence of the Hardy inequality [Ekholm, Kovařík, Krejčiřík 2008]

6 Ω θ D and Ω 0 D Spectral stability have the same spectrum as a set: σ( Ω θ D ) = σ ess( Ω θ D ) = [E 1, ). E 1 is the threshold energy of ω D Difference: existence of the Hardy inequality [Ekholm, Kovařík, Krejčiřík 2008] Hardy-type inequality Ω θ D E 1 ρ, ρ is a positive function. Operator Ω θ D E 1 is subcritical, Ω 0 D E 1 is critical

7 Layout Heat equation in twisted waveguide Quantum waveguides Hardy inequalities Heat equation on Ω θ Asymptotic behaviour Decay rate Self-similarity transformation Numerical solution Numerical methods Comparison The twisted case Conclusion

8 Heat equation on Ω θ u t (x, t) u(x, t) = 0 subject to Dirichlet boundary conditions on Ω θ. From the semigroup theory: u 0 L 2 (Ω θ ) exists uniquely determined generalized solution of the heat equation in the form u(x, t) = e Ω θ D t u 0 (x), where e Ω θ D t : L 2 (Ω θ ) L 2 (Ω θ ) is the semigroup operator associated with the Laplacian Ω θ D. It follows from the spectral mapping theorem that e Ω θ D t L 2 (Ω θ ) L 2 (Ω θ ) = e E 1t.

9 Layout Heat equation in twisted waveguide Quantum waveguides Hardy inequalities Heat equation on Ω θ Asymptotic behaviour Decay rate Self-similarity transformation Numerical solution Numerical methods Comparison The twisted case Conclusion

10 Decay rate We are interested in an additional (polynomial) decay of the semigroup, which will follow if we restrict the class of initial data to the weighted space L 2 (Ω θ, K), where K(x) := e x2 1 /4. Let us define the decay rate: { γ(ω θ ) := sup γ C γ > 0, t 0, e ( θ+e 1 )t K C γ (1 + t) γ}, where. K : L 2 (Ω θ, K) L 2 (Ω θ ). Theorem We have γ(ω θ ) = 1/4 if Ω θ is untwisted, while γ(ω θ ) = 3/4 if Ω θ is twisted. [D. Krejčiřík and E. Zuazua, 2011]

11 Layout Heat equation in twisted waveguide Quantum waveguides Hardy inequalities Heat equation on Ω θ Asymptotic behaviour Decay rate Self-similarity transformation Numerical solution Numerical methods Comparison The twisted case Conclusion

12 Self-similarity transformation Motivation: The Dirichlet Laplacian converges in the norm resolvent sense to one dimensional Schrödinger operator whose potential holds the information about twisting [Šediváková, Krejčiřík, 2011]. One-dimensional heat equation: u t u xx + V (x)u = 0, with V = C ω θ 2. Self-similarity transformation u(x, t) = (t + 1) 1/4 w(y, s), y := (t + 1) 1/2 x, s := ln (t + 1).

13 Using self-similarity transformation we arrive at: Finally, obeying w s 1 2 yw y 1 4 w w yy + e s V (e s/2 y)w = 0. z(y, s) := e y 2 /8 w(y, s), we get expression in (y, s) Ω 0 (0, ): z s z yy + y 2 16 z + es V (e s/2 y)z = 0 This is a parabolic equation with time-dependent coefficients. However, this form is advantageous because of the compactness of the resolvent. Self-similarity transformation is unitary: u(t) = w(s). According to theorem, we expect V = 0, u(t) t 1/4, V 0, u(t) t 3/4.

14 Layout Heat equation in twisted waveguide Quantum waveguides Hardy inequalities Heat equation on Ω θ Asymptotic behaviour Decay rate Self-similarity transformation Numerical solution Numerical methods Comparison The twisted case Conclusion

15 Exact solution for V = 0 Semigroup operator of the heat equation in an integral operator u(x, t) = G(x, y, t)u 0 (y)dy, where the heat kernel is defined as R G(x, y, t) = x y 2 e 4t. 4πt We are able to find numerical solution (Wolfram Mathematica 7.0).

16 Expansion to oscillator basis The first method uses the expansion to the eigenbasis of the harmonic oscillator: u(x, t) = a n (t)ψ n (x). n=1 ψ satisfies the Helmoltz equality ψ + x2 16ψ = λψ, we know explicitly the eigenvalues and -vectors from quantum mechanics: λ n = 1 ( n + 1 ) ( x ), ψ n (x) = N n H n e x Plugging into the heat equation: a(t) = e Mt a(0), where M mn = λ n δ mn ψ m, x2 16 ψ n.

17 Self-similarity solution The solution of self-similarity-transformed equation z s z yy + y 2 16z = 0 may be found again as expansion to harmonic oscillator eigenbasis. Then a(t) = e Mt a(0), where M mn = λ n δ mn. Finally, we apply backward self-similarity transformation. The decay rate function is defined q(t) := ln u(t) ln (1 + t).

18 Layout Heat equation in twisted waveguide Quantum waveguides Hardy inequalities Heat equation on Ω θ Asymptotic behaviour Decay rate Self-similarity transformation Numerical solution Numerical methods Comparison The twisted case Conclusion

19 Time evolution of the norm u : Comparison Decay rate q(t):

20 Layout Heat equation in twisted waveguide Quantum waveguides Hardy inequalities Heat equation on Ω θ Asymptotic behaviour Decay rate Self-similarity transformation Numerical solution Numerical methods Comparison The twisted case Conclusion

21 The twisted waveguide The potential increases the decay rate about 1/2. Analytically, the mathematical background lies in scalling which is more singular than the Dirac s delta interaction and thus leads to the Dirichlet condition at the origin. This problem is already explicitely solvable: the first eigenvalue is 3/4, which coincides with the second eigenvalue of the harmonic oscillator without Dirichlet condition. Decay rate q(t):

22 Conclusion Aim: support the data given by Krejčiřík and Zuazua; show that the decay rate posseses increase about 1/2 in the twisted waveguide in comparison to the untwisted case. Possible extensions: Computation for non-approximated 3D waveguides.

23 Admiration of the audience THANK YOU FOR YOUR ATTENTION