Semra Demirel
Semra Demirel
Semra Demirel
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
Spectral inequalities for Quantum Graphs<br />
<strong>Semra</strong> <strong>Demirel</strong><br />
University of Stuttgart<br />
joint work with Evans M. Harrell, II<br />
”On semiclassical and universal inequalities for eigenvalues of<br />
Quantum Graphs”, Rev. Math. Phys.<br />
Queen Dido Conference, 2010<br />
Idea of the proof<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
Metric Graphs, Metric Trees<br />
✬✩<br />
�<br />
�✫✪<br />
�<br />
❅<br />
�<br />
�<br />
❅<br />
��<br />
� � �<br />
�<br />
◮ V . . . set of vertices<br />
◮ E . . . set of edges, i.e. intervals<br />
o�<br />
Γ<br />
�✏ ✏✏✏✏✏<br />
�<br />
������ �✏ ✏<br />
�<br />
❅ �<br />
�� ◮ metric defined by the distance, e.g. |x| = dist(o, x)<br />
Idea of the proof<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
Quantum Graphs<br />
Idea of the proof<br />
Define Schrödinger operator H(α) in L2 (Γ) = �<br />
e∈E L2 (e), acting<br />
as<br />
H(α)φ = − d 2φ + V (x)φ on every edge e,<br />
dx 2<br />
where φ ∈ D(H(α)) ⇔<br />
φ ∈ H 2 (e) ∀ e ∈ E, �<br />
e∈E<br />
�φ� 2 H 2 (e)<br />
< ∞<br />
with Kirchhoff (Neumann) b.c. in every vertex v ∈ V, i.e. φ is<br />
continuous in v and<br />
�<br />
e∈Ev<br />
dφ<br />
(v) = 0 ∀ v ∈ V.<br />
dx<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
Quantum Graphs<br />
assumption:<br />
then<br />
and<br />
lim V (x) = 0,<br />
|x|→∞<br />
σess(H(α)) = [0, ∞)<br />
σd(H(α)) = {−Ej(α) < 0, j ∈ N}<br />
How do the eigenvalues −Ej(α) depend on V ?<br />
⇓<br />
spectral inequalities: �<br />
j<br />
γ<br />
Ej (α) ≤ ?<br />
Idea of the proof<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
Lieb-Thirring inequalities for metric trees<br />
Theorem [D., Harrell]<br />
Let Γ be a metric tree with a finite number of vertices and edges<br />
and V ∈ Lγ+1/2 (Γ). Then the Lieb-Thirring inequality (LTI)<br />
�<br />
�<br />
1/2<br />
α (V−(x)) γ+1/2 dx<br />
j<br />
E γ<br />
j (α) ≤ Lcl γ,1<br />
holds for all α > 0 and γ ≥ 2.<br />
Lcl γ,1 . . . classical constant.<br />
Γ<br />
Idea of the proof<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
Classical Lieb-Thirring inequalities in R d<br />
H(α) = −α∆ + V (x) in L 2 (R d ), α > 0.<br />
Lieb-Thirring inequalities<br />
�<br />
d/2<br />
α<br />
j<br />
E γ<br />
j (α) ≤ Lγ,d<br />
for suitable values of γ (depending on d) and<br />
Weyl’s law:<br />
Lγ,d ≥ L cl<br />
γ,d :=<br />
lim<br />
α→0+ αd/2 � E γ<br />
j<br />
�<br />
R d<br />
(V−(x)) γ+d/2 dx,<br />
Γ(γ + 1)<br />
2 d π d/2 Γ(γ + 1 + d/2)<br />
(α) = Lcl<br />
γ,d<br />
�<br />
R d<br />
(V−(x)) γ+d/2 dx.<br />
Idea of the proof<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
Stubbe’s proof of sharp LT for γ ≥ 2<br />
◮ A trace formula (”sum rule”) of Harrell-Stubbe ’97, for<br />
H = −α∆ + V :<br />
�<br />
γ<br />
Ej (α) − α2γ<br />
d<br />
�<br />
γ−1<br />
Ej (α)�∇φj� 2 = explicit expr. ≤ 0.<br />
◮ < φj, −∆φj >= �∇φj�2 = ∂Ej(α)<br />
∂α<br />
(Feynman-Hellman)<br />
Thus<br />
�<br />
2<br />
Ej (α) ≤ 2α ∂ �<br />
2<br />
Ej (α)<br />
d ∂α<br />
∂<br />
�<br />
α<br />
∂α<br />
d/2 � E 2<br />
�<br />
j (α) ≤ 0.<br />
Idea of the proof<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
Stubbe’s proof of sharp LTI for γ ≥ 2<br />
∂<br />
�<br />
α<br />
∂α<br />
d/2 � E 2<br />
�<br />
j (α) ≤ 0.<br />
Idea of the proof<br />
And classical LTI is an immediate consequence for γ ≥ 2!<br />
�<br />
(V−(x)) γ+d/2 dx.<br />
α d/2 � E γ<br />
j (α) ≤ lim<br />
α→0+ αd/2 � E γ<br />
j<br />
(α) = Lcl<br />
γ,d<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs<br />
R d
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
Stubbe’s proof of sharp LTI for γ ≥ 2<br />
∂<br />
�<br />
α<br />
∂α<br />
d/2 � E 2<br />
�<br />
j (α) ≤ 0.<br />
Idea of the proof<br />
And classical LTI is an immediate consequence for γ ≥ 2!<br />
�<br />
(V−(x)) γ+d/2 dx.<br />
α d/2 � E γ<br />
j (α) ≤ lim<br />
α→0+ αd/2 � E γ<br />
j<br />
(α) = Lcl<br />
γ,d<br />
Does an analog of the LTI (d = 1) hold also for QGraphs with<br />
the same sharp constants L cl<br />
γ,1 ?<br />
in general: NO!<br />
✛✘<br />
✚✙<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs<br />
R d
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
What about QGraphs?<br />
For metric trees:<br />
�<br />
γ<br />
Ej (α) ≤ Cγ<br />
�<br />
Γ<br />
(V−(x)) γ+1/2 dx, γ ≥ 1/2 [E, F, K]<br />
Cγ = L cl<br />
γ,1?<br />
Idea of the proof<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
What about QGraphs?<br />
For metric trees:<br />
�<br />
γ<br />
Ej (α) ≤ Cγ<br />
�<br />
Γ<br />
(V−(x)) γ+1/2 dx, γ ≥ 1/2 [E, F, K]<br />
Cγ = L cl<br />
γ,1?<br />
YES for γ ≥ 2<br />
Theorem [D., Harrell]<br />
Let Γ be a metric tree with a finite number of vertices and edges<br />
and V ∈ Lγ+1/2 (Γ). Then the Lieb-Thirring inequality (LTI)<br />
�<br />
1/2<br />
α<br />
j<br />
E γ<br />
j (α) ≤ Lcl γ,1<br />
holds for all α > 0 and γ ≥ 2.<br />
�<br />
Γ<br />
(V−(x)) γ+1/2 dx<br />
Idea of the proof<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs
Quantum Graphs Lieb-Thirring inequalities for metric trees Classical LTI in R d<br />
Idea of proof:<br />
◮ sum rule of Harrell and Stubbe:<br />
1<br />
2<br />
� E 2<br />
j < [G, [H, G]]φj, φj > −Ej�[H, G]φj� 2 ≤ 0,<br />
Gφj ∈ D(HΓ)?<br />
◮ averaging argument:<br />
family of piecewise affine functions G<br />
⇒ reduction of the problem to a combinatorial problem<br />
◮ Stubbe’s monotonicity argument<br />
Idea of the proof<br />
<strong>Semra</strong> <strong>Demirel</strong> Queen Dido Conference, 24 May 2010<br />
Spectral inequalities for Quantum Graphs