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Stability of Eigenvalues of Quantum Graphs with Respect to Magnetic Perturbation

Tracy Weyand

Texas A&M University College Station, TX 77843-3368 www.math.tamu.edu/˜tweyand tweyand@math.tamu.edu arXiv:1212.4475, Phil Trans Roy Soc A (joint with G. Berkolaiko)

Texas Analysis and Mathematical Physics Symposium, 2013

Metric Graphs

Γ = {V , E, L} Compact

Weyand Eigenvalues of Quantum Graphs

Metric Graphs

Γ = {V , E, L} Compact Functions: H2(Γ) = ⊕e∈EH2(e) 1st Betti # = |E| − |V | + 1

Weyand Eigenvalues of Quantum Graphs

Quantum Graphs

Metric Graph + Differential Operator Schr¨

• dinger Operator

H0(Γ) : f → − d2

dx2 f (x) + q(x)f (x),

f ∈ H2(Γ, C) f (x) is continuous at v,

• e∈Ev

d dxe f (x)

• v = χvf (v),

χv ∈ R

Weyand Eigenvalues of Quantum Graphs

Quantum Graphs

Metric Graph + Differential Operator Schr¨

• dinger Operator

H0(Γ) : f → − d2

dx2 f (x) + q(x)f (x),

f ∈ H2(Γ, C) f (x) is continuous at v,

• e∈Ev

d dxe f (x)

• v = χvf (v),

χv ∈ R Magnetic Schr¨

• dinger Operator

HA(Γ) : f → − d dx − iA(x) 2 f (x) + q(x)f (x), f ∈ H2(Γ, C) f (x) is continuous at v,

• e∈Ev
• d

dxe − iAe(x)

• f (x)
• v = χvf (v),

χv ∈ R

Weyand Eigenvalues of Quantum Graphs

Magnetic Flux

c1

+

c1

• c2

+

c2

• c1

c2 c c

2 1

αj = c+

j

c−

j

A(x)dx mod 2π Magnetic Flux: α = (α1, α2, . . . , αβ)

Weyand Eigenvalues of Quantum Graphs

Unitarily Equivalent Operators

HA(Γ) : f → − d dx − iA(x) 2 f (x) + q(x)f (x), f ∈ H2(Γ, C) f (x) is continuous at v,

• e∈Ev
• d

dxe − iAe(x)

• f (x)
• v = χvf (v),

χv ∈ R

Weyand Eigenvalues of Quantum Graphs

Unitarily Equivalent Operators

HA(Γ) : f → − d dx − iA(x) 2 f (x) + q(x)f (x), f ∈ H2(Γ, C) f (x) is continuous at v,

• e∈Ev
• d

dxe − iAe(x)

• f (x)
• v = χvf (v),

χv ∈ R Hα(Γ) : f → − d2

dx2 f (x) + q(x)f (x),

f ∈ H2(T, C)        f (x) is continuous at v

• e∈Ev

df dxe (v) = χvf (v)

for v ∈ Γ f (c−

j ) = eiαjf (c+ j )

f ′(c−

j ) = −eiαjf ′(c+ j )

Now we consider λn(α) as a function of α.

Weyand Eigenvalues of Quantum Graphs

Nodal Surplus

φn = # of zeros of the nth eigenfunction νn = # of subgraphs formed by removing the φn zeros from Γ Nodal Surplus: φn − (n − 1) Nodal Deficiency: n − νn

Φ ν Φ ν Φ ν

Φ ν Φ ν Φ ν

Weyand Eigenvalues of Quantum Graphs

Nodal Surplus

φn = # of zeros of the nth eigenfunction νn = # of subgraphs formed by removing the φn zeros from Γ Nodal Surplus: φn − (n − 1) Nodal Deficiency: n − νn

n = 1, Φ1 = 0, ν1=1 n = 2 ,Φ2 = 1, ν2= 2 n = 3 ,Φ3 = 2, ν3= 3

Φ ν Φ ν Φ ν

Weyand Eigenvalues of Quantum Graphs

Nodal Surplus

φn = # of zeros of the nth eigenfunction νn = # of subgraphs formed by removing the φn zeros from Γ Nodal Surplus: φn − (n − 1) Nodal Deficiency: n − νn

n = 1, Φ1 = 0, ν1=1 n = 2 ,Φ2 = 1, ν2= 2 n = 3 ,Φ3 = 2, ν3= 3

n = 1, Φ1 = 0, ν1 = 1 n = 2, Φ2 = 1, ν2 = 2 n = 3, Φ3 = 2, ν3 = 3

Weyand Eigenvalues of Quantum Graphs

Nodal Surplus

φn = # of zeros of the nth eigenfunction νn = # of subgraphs formed by removing the φn zeros from Γ Nodal Surplus: φn − (n − 1) Nodal Deficiency: n − νn

n = 1, Φ1 = 0, ν1=1 n = 2 ,Φ2 = 1, ν2= 2 n = 3 ,Φ3 = 2, ν3= 3 n = 1, Φ1 = 0, ν1 = 1 n = 2, Φ2 = 2, ν2 = 2 n = 3, Φ3 = 2, ν3 = 3

Weyand Eigenvalues of Quantum Graphs

Morse Index

Morse Index = # of negative eigenvalues of the Hessian matrix Hi,j = d2λn(α) dαidαj

−1 −0.5 0.5 1 −1 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1 −0.5 0.5 1 −1 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2

Morse Index = 0 Morse Index = 1 Morse Index = 2

Weyand Eigenvalues of Quantum Graphs

Main Result

Theorem (Berkolaiko & Weyand, 2013) Let λn be a simple eigenvalue of H0 whose eigenfunction has φ internal zeros. Consider the eigenvalues λn(α) of Hα as a function of α: α = (0, 0, . . . , 0) is a non-degenerate critical point of λn(α) and the Morse index of this critical point is equal to φ − (n − 1)

Weyand Eigenvalues of Quantum Graphs

Partitions

Proper m-Partition: Set of m points, none of which lie on vertices Partition Subgraphs: Subgraphs Γj formed by applying Dirichlet conditions at the m-partition points

Weyand Eigenvalues of Quantum Graphs

Corollary

Λ(P) := maxjλ1(Γj) Equipartition: All partition subgraphs have the same first eigenvalue

Weyand Eigenvalues of Quantum Graphs

Corollary

Λ(P) := maxjλ1(Γj) Equipartition: All partition subgraphs have the same first eigenvalue Corollary (Berkolaiko & Weyand, 2013) Consider Λ on the set of equipartitions: the φ-equipartition formed from the zeros of the nth eigenfunction is a non-degenerate critical point of Λ and the Morse index of this critical point is equal to n − ν. Note: This strengthens the result of Band, Berkolaiko, Raz, and Smilansky (‘12)

Weyand Eigenvalues of Quantum Graphs

Can one “hear” the shape of a graph?

Given only eigenvalues, can one reconstruct the graph?

Weyand Eigenvalues of Quantum Graphs

Can one “hear” the shape of a graph?

Given only eigenvalues, can one reconstruct the graph? No, isospectral quantum graphs exist (Sunada, ’85). Cannot Determine: (Band and Parzanchevski, ’10) # of edges and vertices # of independent cycles (β = |E| − |V | + 1)

2b 2c a a 2a c c b b

Weyand Eigenvalues of Quantum Graphs

Only a Tree is a Tree

On a tree, φn = n − 1 ∀n. Theorem (Band, 2013) If φn = n − 1 ∀n, then the graph is a tree.

Weyand Eigenvalues of Quantum Graphs

References

• R. BAND, The nodal count {0, 1, 2, 3, . . .} is a tree. preprint

arXiv:1212.6710 [math-ph], 2012.

• R. BAND, G. BERKOLAIKO, H. RAZ, AND U. SMILANSKY, The

number of nodal domains on quantum graphs as a stability index of graph partitions, Comm. Math. Phys., 311 (2012), pp. 815-838.

• G. BERKOLAIKO AND T. WEYAND, Stability of eigenvalues of

quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions, Philosophical Transactions of the Royal Society A, accepted arXiv:1212.4475 [math-ph], 2012.

Contact Information

Tracy Weyand www.math.tamu.edu/˜tweyand tweyand@math.tamu.edu

Weyand Eigenvalues of Quantum Graphs