Optimal potentials on quantum graphs with -couplings Andrea Serio - - PowerPoint PPT Presentation
Introduction Previous works and main results Methods Conclusion Optimal potentials on quantum graphs with -couplings Andrea Serio joint work with Pavel Kurasov Differential Operators on Graphs and Waveguides February 25 - March 1, 2019
Introduction Previous works and main results Methods Conclusion
joint work with Pavel Kurasov
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
dx2 + q(x) with q ∈ L1
◮ The problem was originally formally posed by A. G. Ramm in
◮ The case of Dirichlet boundary condition on the interval has
◮ The case with fixed Robin conditions on the interval were
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
◮ Schr¨
q (Γ) = − d2 dx2 + q(x),
q ∈ L1 in the Hilbert space L2(Γ).
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
◮ Schr¨
q (Γ) = − d2 dx2 + q(x),
q ∈ L1 in the Hilbert space L2(Γ).
◮ We consider delta vertex conditions (δ-v.c.) at the vertices v ∈ V
where ∂ψ(xj) denotes the normal derivative of ψ. Hence Lh
q (Γ) acts on the following domain
D Lh
q (Γ)
:=
N
W1
2(en)
− d2 dx2 u|en + qu|en ∈ L2(en), ∀n;
∂u(xi) = h(v)u(v) ∀v ∈ V (Γ)
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
◮ Schr¨
q (Γ) = − d2 dx2 + q(x),
q ∈ L1 in the Hilbert space L2(Γ).
◮ We consider delta vertex conditions (δ-v.c.) at the vertices v ∈ V
where ∂ψ(xj) denotes the normal derivative of ψ. Hence Lh
q (Γ) acts on the following domain
D Lh
q (Γ)
:=
N
W1
2(en)
− d2 dx2 u|en + qu|en ∈ L2(en), ∀n;
∂u(xi) = h(v)u(v) ∀v ∈ V (Γ)
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q(Γ) with L1-potential
q(Γ)) = {λ1 ≤ λ2 ≤ λ3 ≤ . . . }.
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q(Γ) with L1-potential
q(Γ)) = {λ1 ≤ λ2 ≤ λ3 ≤ . . . }.
q(Γ)
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q(Γ) with L1-potential
q(Γ)) = {λ1 ≤ λ2 ≤ λ3 ≤ . . . }.
q(Γ)
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
◮ Eigenvalues inequalities for the Laplacian on metric graphs (q ≡ 0). Kennedy, Kurasov, Malenova and Mugnolo; Band and Levy; Rohleder; Ariturk; Berkolaiko, Kennedy, Kurasov and Mugnolo; Kurasov, S. ◮ Lower bound of the Ground state of Quantum Graphs (q ≡ 0)
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
◮ Eigenvalues inequalities for the Laplacian on metric graphs (q ≡ 0). Kennedy, Kurasov, Malenova and Mugnolo; Band and Levy; Rohleder; Ariturk; Berkolaiko, Kennedy, Kurasov and Mugnolo; Kurasov, S. ◮ Lower bound of the Ground state of Quantum Graphs (q ≡ 0)
q(Γ) be a Schr¨
q− + h− , and
q+ + h+. Then
0([0, L])) ≤ λ1(Lh q(Γ))
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
◮ Eigenvalues inequalities for the Laplacian on metric graphs (q ≡ 0). Kennedy, Kurasov, Malenova and Mugnolo; Band and Levy; Rohleder; Ariturk; Berkolaiko, Kennedy, Kurasov and Mugnolo; Kurasov, S. ◮ Lower bound of the Ground state of Quantum Graphs (q ≡ 0)
q(Γ) be a Schr¨
q− + h− , and
q+ + h+. Then
0([0, L])) ≤ λ1(Lh q(Γ))
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
(q,h) Under ass. 1,2,3.
q(Γ)
1.
Γ q(x) dx = Q, (the total strength of the potential be fixed),
2.
h(v) = H, (the total strength of the singular interaction be fixed),
if Q ≥ 0, q(x) ≤ 0 if Q ≤ 0, ∀x ∈ Γ.
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
◮ The optimisation problem is independent of the topology of
◮ If Q · H ≥ 0, then the optimal configuration (q∗, h∗) exists
◮ If Q · H < 0, then the optimal configuration does not exist,
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q(Γ), then ψ = ψ1,
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q(Γ), then ψ = ψ1,
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
We denote by QLh
q (Γ) the quadratic form associated with Lh
q (Γ), given by
QLh
q (Γ)(u) = u′2 +
q(x)|u(x)|2 dx +
h(v)|u(v)|2, (3)
D
q (Γ)
N
W1
2(en)
Lh
q (Γ)
.
q(Γ)
u∈D
q (Γ)
q (Γ)(u)
The minimiser coincides with the ground state ψ
Lh
q (Γ)
1
( = ψq
1 when h is fixed) Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
We denote by QLh
q (Γ) the quadratic form associated with Lh
q (Γ), given by
QLh
q (Γ)(u) = u′2 +
q(x)|u(x)|2 dx +
h(v)|u(v)|2, (3)
D
q (Γ)
N
W1
2(en)
Lh
q (Γ)
.
q(Γ)
u∈D
q (Γ)
q (Γ)(u)
The minimiser coincides with the ground state ψ
Lh
q (Γ)
1
( = ψq
1 when h is fixed) Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Let the delta couplings (h) be fixed. Assume that q∗ ≥ 0 such that supp q∗ ⊆ Mq∗ =
1 (x) = max y∈Γ ψq∗ 1 (y)
(4) then ◮ the potential q∗ is optimal in the sense, that λ1(Lh
q∗(Γ)) ≥ λ1(Lh q (Γ))
(5) holds ∀q ≥ 0 with the same total strength
Γ q∗(x)dx = Γ q(x)dx.
◮ The optimal potential q∗ is unique.
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q(Γ)
q∗(Γ)
q∗) = Qh q∗(Γ, ψq∗ 1 )
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q(Γ)
q∗(Γ)
q∗) = Qh q∗(Γ, ψq∗ 1 )
1 2 2 +
1|2 dx +
1|2(v)
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q(Γ)
q∗(Γ)
q∗) = Qh q∗(Γ, ψq∗ 1 ) ≥ Qh q(Γ, ψq∗ 1 )
1 2 2 + Q sup Γ
1|2 +
1|2(v)
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q(Γ)
q∗(Γ)
q∗) = Qh q∗(Γ, ψq∗ 1 ) ≥ Qh q(Γ, ψq∗ 1 ) ≥ min ψ=1Qh q(Γ, ψ) = λ1(Lh q).
1 2 2 +
1|2 dx +
1|2(v)
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q(Γ)
q∗(Γ)
q∗) = Qh q∗(Γ, ψq∗ 1 ) ≥ Qh q(Γ, ψq∗ 1 ) ≥ min ψ=1Qh q(Γ, ψ) = λ1(Lh q).
ψ=1
2 +
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
On the loop graph L = [− ℓ
2 , + ℓ 2 ] with H, Q ≥ 0 there exists an optimal configuration
(q∗, h∗) q∗(x) =
ψ∗(x) =
|x| − ℓ
2 + α
) |x| ≥ ℓ
2 − α
1 |x| < ℓ
2 − α
where α and k are the smallest positive solutions of
2k tan(kα) = H
0.0 0.5 1.0 1.5 0.5 1.0 1.5
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
◮ Qn > 0 then qα,k(x) =
k2, ψα,k(x) =
k |x| − α − ℓn
2
|x| ≥ α − ℓn
2 ;
1, |x| ≤ α − ℓn
2 ,
(7) ◮ Qn = 0 then qα,k(x) ≡ 0, ψα,k(x) = cos kα cos kℓn/2 cos kx, (8) ◮ Hn = 2k tan kℓn/2. where α is a common real parameter. We calculate the corresponding total potential and total singular interaction and put them equal to Q and H respectively and get the following system
k2
ℓn≥2α
(ℓn − 2α) = Q,
2k tan kα +
2k tan kℓn/2 = H. (9)
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Any flower graph F, for any given H, Q ≥ 0 total strength of the interactions, admits an optimal potential q with corresponding optimal ground state ψ > 0 such that
q (F)
= Λ(F, H, Q),
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Let ψ ∈ D (Lh
q (Γ)) and let v ∈ V (Γ), v1, v2 ∈ V (Γc), v = v1 ∪ v2.
We define the restriction of the δ-vertex conditions h to hc on Γc in the following way hc(w) :=
if w ∈ V (Γc) \ {v1, v2},
1 ψ(v)
if w = v1, v2. (10) therefore we say that the graph Γ is cut in v into Γc with respect to ψ.
v
1
2
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Let ψ ∈ D (Lh
q (Γ)) and let v ∈ V (Γ), v1, v2 ∈ V (Γc), v = v1 ∪ v2.
We define the restriction of the δ-vertex conditions h to hc on Γc in the following way hc(w) :=
if w ∈ V (Γc) \ {v1, v2},
1 ψ(v)
if w = v1, v2. (10) therefore we say that the graph Γ is cut in v into Γc with respect to ψ.
Let vi, vj ∈ V (Γ). We define the new δ-vertex conditions hg on Γg in the following way hg(w) :=
if w ∈ V (Γc) \ {v1, v2}, h(v1) + h(v2) if w = v1 ∪ v2. (11) therefore we say that the graph Γg is obtained by glueing vi, vj ∈ V (Γ) into v.
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
λ1
q (Γ)
= λ1
q (Γc)
Apply the corollary of Perron-Frobenius Theorem on Γc and ψ.
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
λ1
q (Γ)
= λ1
q (Γc)
Apply the corollary of Perron-Frobenius Theorem on Γc and ψ.
Λ (Γ, H, Q) ≤ Λ (Γg, H, Q)
Notice that by the definition of glueing C (Γg) C (Γ); Hence by glueing the domain shrinks D g := D
hg q
(Γg)
q (Γ)
Qhg
q (Γg, ψ) = Qh q (Γ, ψ), ∀ψ ∈ D g. Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
λ1
q (Γ)
= λ1
q (Γc)
Apply the corollary of Perron-Frobenius Theorem on Γc and ψ.
Λ (Γ, H, Q) ≤ Λ (Γg, H, Q)
Notice that by the definition of glueing C (Γg) C (Γ); Hence by glueing the domain shrinks D g := D
hg q
(Γg)
q (Γ)
Qhg
q (Γg, ψ) = Qh q (Γ, ψ), ∀ψ ∈ D g.The inequality follows
from comparing the minimum of the Rayleigh quotient over the domains D g, D
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
λ1
q (Γ)
= λ1
q (Γc)
Apply the corollary of Perron-Frobenius Theorem on Γc and ψ.
Λ (Γ, H, Q) ≤ Λ (Γg, H, Q)
Notice that by the definition of glueing C (Γg) C (Γ); Hence by glueing the domain shrinks D g := D
hg q
(Γg)
q (Γ)
Qhg
q (Γg, ψ) = Qh q (Γ, ψ), ∀ψ ∈ D g.The inequality follows
from comparing the minimum of the Rayleigh quotient over the domains D g, D
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
λ1
q (Γ)
= λ1
q (Γc)
Apply the corollary of Perron-Frobenius Theorem on Γc and ψ.
Λ (Γ, H, Q) ≤ Λ (Γg, H, Q)
Notice that by the definition of glueing C (Γg) C (Γ); Hence by glueing the domain shrinks D g := D
hg q
(Γg)
q (Γ)
Qhg
q (Γg, ψ) = Qh q (Γ, ψ), ∀ψ ∈ D g.The inequality follows
from comparing the minimum of the Rayleigh quotient over the domains D g, D
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
λ1
q (Γ)
= λ1
q (Γc)
Apply the corollary of Perron-Frobenius Theorem on Γc and ψ.
Λ (Γ, H, Q) ≤ Λ (Γg, H, Q)
Notice that by the definition of glueing C (Γg) C (Γ); Hence by glueing the domain shrinks D g := D
hg q
(Γg)
q (Γ)
Qhg
q (Γg, ψ) = Qh q (Γ, ψ), ∀ψ ∈ D g.The inequality follows
from comparing the minimum of the Rayleigh quotient over the domains D g, D
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Given Γ, H, Q
◮ Consider FΓ, the flower graph with the same edges as Γ all joined in
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Given Γ, H, Q
◮ Consider FΓ, the flower graph with the same edges as Γ all joined in
◮ Consider either
◮ the optimal potential q and optimal GS ψ of LH
q (FΓ), or...
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Given Γ, H, Q
◮ Consider FΓ, the flower graph with the same edges as Γ all joined in
◮ Consider either
◮ the optimal potential q and optimal GS ψ of LH
q (FΓ), or...
◮ an optimising sequence qn, ψn.
◮ Cut FΓ along ψ (resp. ψn) finitely times so to get obtain Γ...
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Given Γ, H, Q
◮ Consider FΓ, the flower graph with the same edges as Γ all joined in
◮ Consider either
◮ the optimal potential q and optimal GS ψ of LH
q (FΓ), or...
◮ an optimising sequence qn, ψn.
◮ Cut FΓ along ψ (resp. ψn) finitely times so to get obtain Γ... ...at
each cut ensures that
◮ the function ψ (resp. ψn) is still the GS by the corollary of the
Perron-Frobenius Theorem,
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Given Γ, H, Q
◮ Consider FΓ, the flower graph with the same edges as Γ all joined in
◮ Consider either
◮ the optimal potential q and optimal GS ψ of LH
q (FΓ), or...
◮ an optimising sequence qn, ψn.
◮ Cut FΓ along ψ (resp. ψn) finitely times so to get obtain Γ... ...at
each cut ensures that
◮ the function ψ (resp. ψn) is still the GS by the corollary of the
Perron-Frobenius Theorem,
◮ the pair q, hc is still an optimal on the cut graph. Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Given Γ, H, Q
◮ Consider FΓ, the flower graph with the same edges as Γ all joined in
◮ Consider either
◮ the optimal potential q and optimal GS ψ of LH
q (FΓ), or...
◮ an optimising sequence qn, ψn.
◮ Cut FΓ along ψ (resp. ψn) finitely times so to get obtain Γ... ...at
each cut ensures that
◮ the function ψ (resp. ψn) is still the GS by the corollary of the
Perron-Frobenius Theorem,
◮ the pair q, hc is still an optimal on the cut graph.
◮ Λ(Γ, H, Q) ≤ Λ(FΓ, H, Q) = λ(LH
q (FΓ)) = λ(Lhc q (Γ)).
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Given Γ, H, Q
◮ Consider FΓ, the flower graph with the same edges as Γ all joined in
◮ Consider either
◮ the optimal potential q and optimal GS ψ of LH
q (FΓ), or...
◮ an optimising sequence qn, ψn.
◮ Cut FΓ along ψ (resp. ψn) finitely times so to get obtain Γ... ...at
each cut ensures that
◮ the function ψ (resp. ψn) is still the GS by the corollary of the
Perron-Frobenius Theorem,
◮ the pair q, hc is still an optimal on the cut graph.
◮ Λ(Γ, H, Q) ≤ Λ(FΓ, H, Q) = λ(LH
q (FΓ)) = λ(Lhc q (Γ)).
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q∗(Γ).
Lh′
q∗(Γ)
q∗(FΓ)
Lh′
n
qn (Γ)
qn(FΓ)
Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
q∗(Γ).
Lh′
q∗(Γ)
q∗(FΓ)
Lh′
n
qn (Γ)
qn(FΓ)
Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
The optimal configuration has the following properties: ◮ The optimal configuration does not depend
lengths of its edges,
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
The optimal configuration has the following properties: ◮ The optimal configuration does not depend
lengths of its edges, ◮ The optimal ground state ψ attains the same value at all the vertices ψ(v) = C, ∀v ∈ V , ◮ Moreover, if the potential is dominant over the singular interaction then Λ(Γ, H, Q) can be computed just from
◮ L(Γ) the total length of the graph, ◮ ♯E(Γ) the number of edges, ◮ H, Q the strength of the interactions. Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
The optimal configuration has the following properties: ◮ The optimal configuration does not depend
lengths of its edges, ◮ The optimal ground state ψ attains the same value at all the vertices ψ(v) = C, ∀v ∈ V , ◮ Moreover, if the potential is dominant over the singular interaction then Λ(Γ, H, Q) can be computed just from
◮ L(Γ) the total length of the graph, ◮ ♯E(Γ) the number of edges, ◮ H, Q the strength of the interactions. Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
The optimal configuration has the following properties: ◮ The optimal configuration does not depend
lengths of its edges, ◮ The optimal ground state ψ attains the same value at all the vertices ψ(v) = C, ∀v ∈ V , ◮ Moreover, if the potential is dominant over the singular interaction then Λ(Γ, H, Q) can be computed just from
◮ L(Γ) the total length of the graph, ◮ ♯E(Γ) the number of edges, ◮ H, Q the strength of the interactions. Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Andrea Serio Optimal potentials on quantum graphs with δ-couplings
Introduction Previous works and main results Methods Conclusion
Andrea Serio Optimal potentials on quantum graphs with δ-couplings