Surgery of quantum graphs: eigenvalues and heat kernels · 2018-03-26 · Spectral inequalities...

Surgery ofquantum graphs:eigenvalues and

heat kernels

Delio Mugnolo

DiscreteLaplacians

Quantum graphLaplacians

Spectralinequalities

Heat kernels

Surgery of quantum graphs:eigenvalues and heat kernels

Delio Mugnolo

University of Hagen

March 22, 2018

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Heat kernels

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Braess’ paradox

In unfavorable cases, an extension of the roadnetwork may result in longer running times.

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Braess’ paradox

In unfavorable cases, an extension of the roadnetwork may result in longer running times.

Figure: D. Braess 1968

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Braess’ metatheorem

Networks don’t necessarily become more efficient bymerely getting denser and denser!

In the context of quantum graphs:

▸ Exner-Jex 2012

▸ Kurasov-Malenova-Naboko 2013

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Outline

▸ Discrete graphs

▸ Quantum graphs

▸ Heat kernels

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Let G = (V,E) be a loopless (finite) graph (possibly amultigraph, no loops!).Regard f ∶ V → C as a function on G: the natural Hilbertspace is `2(V) ≃ CV.

Upon chosing an arbitrary orientation, consider the incidencematrix I = (ιve):

ιve ∶=⎧⎪⎪⎪⎨⎪⎪⎪⎩

−1 if v is initial endpoint of e,+1 if v is terminal endpoint of e,0 otherwise

ThenLG ∶= IIT

defines a (bounded) self-adjoint, positive semidefiniteoperator on `2(V): the discrete Laplacian (Kirchhoff 1847,Brooks-Smith-Stone-Tutte 1940, Anderson-Morley 1971).

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Let G = (V,E) be a loopless (finite) graph (possibly amultigraph, no loops!).Regard f ∶ V → C as a function on G: the natural Hilbertspace is `2(V) ≃ CV.

Upon chosing an arbitrary orientation, consider the incidencematrix I = (ιve):

ιve ∶=⎧⎪⎪⎪⎨⎪⎪⎪⎩

−1 if v is initial endpoint of e,+1 if v is terminal endpoint of e,0 otherwise

ThenLG ∶= IIT

defines a (bounded) self-adjoint, positive semidefiniteoperator on `2(V): the discrete Laplacian (Kirchhoff 1847,Brooks-Smith-Stone-Tutte 1940, Anderson-Morley 1971).

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Spectral graph theory

The spectrum {0 = λ0, λ1, . . . , λ∣V∣−1} of LG carries muchinteresting information:

▸ multiplicity of 0 in σ(LG) = # of connectedcomponents of G

▸ the higher the spectral gap

λ1(LG) = inff ∈`2(V)f ⊥1

∥IT f ∥2`2(E)

∥f ∥2`2(V)

the more connected G (λ1(LG) is also called algebraicconnectivity of G).

▸ Eigenfunctions associated with λ1(LG), λ2(LG) inducevery reasonable drawings of planar graphs

▸ . . . (see e.g. Spielman 2007, Spielman 2010)

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“Qualitative” spectral graph theory

Idea: relate spectral quantities of LG with somecombinatorial quantities of G.

Example

▸ Fiedler 1973: η ≥ λ1(LG) ≥ 2η (1 − cos π∣V∣)

▸ Dodziuk 1984, Alon-Milman 1985:2h ≥ λ1(LG) ≥ h2

2 degmax

▸ McKay 1988, Chung 1989, Mohar 1991:degmax(∣V∣−1)(2 D−∣V∣+1)+ ≥ λ1(LG) ≥ 4

∣V∣D

(η edge connectivity, h Cheeger constant of G,degmax maximal degree, D diameter)

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Inserting edges tends to lower λ1(LG), but. . .

Figure: C. Maas 1987

. . . sometimes λ1(LG) = λ1(LG′)

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Quantum graphs

A quantum graph (or metric graph, or network) G isobtained by associating an interval (0, è) of length è witheach edge e of G = (V,E).

Natural Hilbert space on G:

L2(G) ∶=⊕e∈E

L2(0, è)

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No boundary conditions can be imposed on functions inL2(G), so functions in L2(G) do not see thecombinatorics of G.

Introduce

C(G) ∶= {f ∈⊕e∈E

C [0, `e] ∶ f is continuous in each v ∈ V}

and

H1(G) ∶= {f = (fe)e∈E ∈ C(G) ∶ fe ∈ H1(0, `e) ∀e ∈ E}

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No boundary conditions can be imposed on functions inL2(G), so functions in L2(G) do not see thecombinatorics of G.

Introduce

C(G) ∶= {f ∈⊕e∈E

C [0, `e] ∶ f is continuous in each v ∈ V}

and

H1(G) ∶= {f = (fe)e∈E ∈ C(G) ∶ fe ∈ H1(0, `e) ∀e ∈ E}

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Spectral gap of quantum graphs

In analogy with

λ1(LG) = inff ∈`2(V)f ⊥1

∥IT f ∥2`2(E)

∥f ∥2`2(V)

and λ1(∆) = inff ∈H1(0,1)

f ⊥1

∥f ′∥2L2(0,1)

∥f ∥2L2(0,1)

consider

λ1(∆G) ∶= inff ∈H1(G)

f ⊥1

∥f ′∥2L2(G)

∥f ∥2L2(G)

λ1(∆G) is the spectral gap of ∆G , the self-adjoint, positivesemidefinite operator on L2(G) associated with

a(f ) ∶=∑e∈E∫

`e

0∣f ′∣2, f ∈ H1(G)

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Spectral gap of quantum graphs

In analogy with

λ1(LG) = inff ∈`2(V)f ⊥1

∥IT f ∥2`2(E)

∥f ∥2`2(V)

and λ1(∆) = inff ∈H1(0,1)

f ⊥1

∥f ′∥2L2(0,1)

∥f ∥2L2(0,1)

consider

λ1(∆G) ∶= inff ∈H1(G)

f ⊥1

∥f ′∥2L2(G)

∥f ∥2L2(G)

λ1(∆G) is the spectral gap of ∆G , the self-adjoint, positivesemidefinite operator on L2(G) associated with

a(f ) ∶=∑e∈E∫

`e

0∣f ′∣2, f ∈ H1(G)

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A secular equation yielding all eigenvalues of ∆G is known(von Below 1985, Kottos-Smilansky 1997) but is useless inpractice.

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Spectral estimates for quantum graphs

Proposition (Nicaise 1987)

▸ λ1(∆G) ≥ π2

L2

▸ λ1(∆G) ≥ h2

4

Nicaise’ first estimate is sharp: it is attained for path graphs.Also the second estimate can be shown to be sharp, adapting an

example invented by Buser.

(L total length, h Cheeger constant of G)

(Many upper estimates by Kennedy-Kurasov-Malenova-M.2016; Band-Levy 2016; Rohleder 2016; Ariturk 2016)

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Proof of Nicaise’ isoperimetric inequality #1▸ Cut through all cycles of G to turn it into a tree G′:

since H1(G) ⊂ H1(G′), this lowers the Rayleighquotient.

▸ Consider the double cover G′(2) of the tree G′ (i.e.,

double each edge): the total length is 2L.▸ Each f ∈ H1(G) induces f(2) ∈ H1(G′(2)) canonically:

their Rayleigh quotients coincide.▸ For f(2) ∈ H1(G′(2)) define g ∈ H1(G′′) (≡cycle of length 2L)

by suitably dropping a few continuity conditions in thevertices (transverse the graph like in a Eulerian cycle).

v0

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Proof of Nicaise’ isoperimetric inequality #2

▸ If f is an eigenfunction for λ1(∆G), g is still a testfunction for G′′ ⇒ λ1(∆G′′) is a lower bound forλ1(∆G)

▸ λ1(∆G′′) is just the spectral gap (=lowest nonzero

eigenvalue) of the second derivative on (0,2L) with

periodic boundary conditions, i.e. 4π2

(2L)2 .

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Nicaise’ isoperimetric inequality has beenrediscovered/reproved by

▸ Solomyak (2002)

▸ Friedlander (2005)

▸ Berkolaiko-Kennedy-Kurasov-M. (2017)

▸ Berkolaiko-Kennedy-Kurasov-M. (2018)

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Proposition (Kurasov-Naboko 2014, Band-Levy 2016)

Nicaise isoperimetric inequality can be improved to

λ1(∆G) ≥4π2

L2

▸ if G is Eulerian;

▸ if, more generally, G is doubly edge connected.

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Proposition (Kurasov-Naboko 2014, Band-Levy 2016)

Nicaise isoperimetric inequality can be improved to

λ1(∆G) ≥4π2

L2

▸ if G is Eulerian;

▸ if, more generally, G is doubly edge connected.

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Kennedy–Kurasov–Malenova–M. 2016,Kennedy-M. 2016,Berkolaiko–Kennedy–Kurasov–M. 2017

estimates on λ1(∆G) upper lower

total length 7 π2

L2

total length + # edges π2E2

L2 7

diameter + # vertices π2(V+1)2

D2 7

diameter + # edges 4π2E2

D2π2

D2E2

diameter + total length π2(4L−3D)D3

1DL

Cheeger constant π2h2E2

4h2

4

edge connectivity + total length 7 ( ηπL+`max(η−2)+ )

2

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Proposition (Nicaise 1987;Berkolaiko-Kennedy-Kurasov-M 2017)

▸ λ0(∆G) ≥ π2

4L2 if there are Dirichlet vertices.

▸ λ0(∆G) ≥ π2

D2 if G is a tree and all leaves are Dirichlet.

▸ λ0(∆G) ≥ π2

L2 if there are Dirichlet vertices and thegraph is doubly connected.

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Surgery of quantum graphs - #1

Proposition (Berkolaiko-Kennedy-Kurasov-M)

λk(∆G) ≥ λk(∆G′) for all k

if G′ is obtained from G by

▸ cutting through two previously identified vertices;

Gv0 v2

v1 v2

v3

v4 G′

▸ attaching a pendant (e.g., an edge) to one vertex;

▸ lengthening an edge

▸ replacing k parallel edges by one edge of same totallength (“unfolding G”)

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Given any eigenfunction ψ for λ1(∆G), then

λ1(∆G) ≥ λ1(G′)


▸ deleting an edge where ψ vanishes identically;

▸ inserting an edge of length ≥ π√λ1(∆G)

between any v ,w ;

▸ inserting an edge between any v ,w s.t. ψ(v) = ψ(w);

▸ replacing k parallel edges (along which ψ is growing) bym ≤ k equilateral edges of same total length;

G G′

▸ cutting a subgraph H with ψ(H) ⊂ [0, κ] and attachingany graph of same length at vertices where ψ is ≥ κ.

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Given any eigenfunction ψ for λ1(∆G), then

λ1(∆G) ≥ λ1(G′)


▸ deleting an edge where ψ vanishes identically;

▸ inserting an edge of length ≥ π√λ1(∆G)

between any v ,w ;

▸ inserting an edge between any v ,w s.t. ψ(v) = ψ(w);

▸ replacing k parallel edges (along which ψ is growing) bym ≤ k equilateral edges of same total length;

G G′

▸ cutting a subgraph H with ψ(H) ⊂ [0, κ] and attachingany graph of same length at vertices where ψ is ≥ κ.

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Comparison principles for quantum graphs

Proposition (Berkolaiko-Kennedy-Kurasov-M 2017)

▸ The lasso graph with loop length S has lowest λ1(∆G)among all graphs with circumference S .

v− v+

▸ The dumbbell graph with handle length D and loopslength (L −D)/2 has lowest λ1(∆G) among all graphswith total length L and diameter D.

v− v+

▸ The dumbbell graph with handle length L−V and loopslength V /2 has lowest λ1(∆G) among all graphs withdoubly connected part of length V .

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Diffusion on graphs and quantum graphs

Laplacians on graphs and quantum graphs are associatedwith Dirichlet forms:

▸ Beurling-Deny 1959, Keller-Lenz 2010 (discrete graphs)

▸ Kramar Fijavz-M-Sikolya 2007 (quantum graphs)

⇒ e−tLG

and et∆G are submarkovian semigroups.

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Parabolic Braess’ paradox

Because

sup∥f ∥L2=1

∥e−t∆G f − ∫Gf ∥L2 = o(e−tλ1(∆G))

surgery principle can be interpreted for (connected) quantumgraphs: e.g.

▸ By identifying vertices of a quantum graph,convergence to equilibrium becomes faster.

▸ If there exist two vertices v,w and an eigenfunction ψ(on G) for λ1 s.t. ψ(v) = ψ(w), then diffusion onG′ ∶= G ∪ vw converges to equilibrium more slowly thanon G.

(For comparison: diffusion on G′ ∶= G ∪ vw converges toequilibrium more quickly than on G.)

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What about diffusion between pairs of vertices?Diffusing from a to z is faster on

Diffusing from b to c is faster on

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Domination: the right notion?Domination of a semigroup (etA)t≥0 by another positivesemigroup (etB)t≥0 means that

∣etAf ∣ ≤ etB ∣f ∣ for all t ≥ 0and all f ≥ 0;

Examples:

▸ (et∆N )t≥0 dominates (et∆D)t≥0;

▸ (e−t(∆N)2)t≥0 does not dominate (et∆N )t≥0;

Proposition (Ouhabaz 1996)

Let a ∼ A, b ∼ B be Dirichlet forms on H, D(a) ⊂ D(b).TFAE:

▸ etA ≤ etB ;

▸ Re a(u, v) ≥ b(∣u∣, ∣v ∣) for all u ∈ D(a) and v ∈ D(b)such that ∣v ∣ ≤ ∣u∣; and additionally, D(a) is an ideal ofD(b).

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Domination: the right notion?Domination of a semigroup (etA)t≥0 by another positivesemigroup (etB)t≥0 means that

∣etAf ∣ ≤ etB ∣f ∣ for all t ≥ 0and all f ≥ 0;

Examples:

▸ (et∆N )t≥0 dominates (et∆D)t≥0;

▸ (e−t(∆N)2)t≥0 does not dominate (et∆N )t≥0;

Proposition (Ouhabaz 1996)

Let a ∼ A, b ∼ B be Dirichlet forms on H, D(a) ⊂ D(b).TFAE:

▸ etA ≤ etB ;

▸ Re a(u, v) ≥ b(∣u∣, ∣v ∣) for all u ∈ D(a) and v ∈ D(b)such that ∣v ∣ ≤ ∣u∣; and additionally, D(a) is an ideal ofD(b).

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Proposition

▸ If G is subgraph of G′, then e−tLG /≤ e−tL

G′ /≤ e−tLG

▸ If G′ is obtained from G by cutting through twopreviously identified vertices, then

e−tLG /≤ e−tL

G′ /≤ e−tLG

.

In particular, there is no domination (in either direction) fordiffusion on spanning (either combinatorial or quantum)trees.

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Interwoven semigroups

Given f > 0, (etAf )t∈[0,∞), (etB f )t∈[0,∞) are interwoven if forall t ∈ [0,∞) and some t1, t2 ≥ t one has

et1B f > et1Af and et2Af > et2B f .

Theorem (Gluck-M 2018)

Let (X , µ) be a finite measure space and A,B be distinctself-adjoint operators on L2(X , µ). If (etA)t≥0, (etB)t≥0 arepositive and map L2(X ) into L∞(X ), then TFAE:

▸ s(A) = s(B)▸ there exists 0 < f ∈ L2 such that the orbits (etAf )t≥0

and (etB f )t≥0 are interwoven.

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Interwoven semigroups

Given f > 0, (etAf )t∈[0,∞), (etB f )t∈[0,∞) are interwoven if forall t ∈ [0,∞) and some t1, t2 ≥ t one has

et1B f > et1Af and et2Af > et2B f .

Theorem (Gluck-M 2018)

Let (X , µ) be a finite measure space and A,B be distinctself-adjoint operators on L2(X , µ). If (etA)t≥0, (etB)t≥0 arepositive and map L2(X ) into L∞(X ), then TFAE:

▸ s(A) = s(B)▸ there exists 0 < f ∈ L2 such that the orbits (etAf )t≥0

and (etB f )t≥0 are interwoven.

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Proposition

▸ If G,G′ are two graphs on V, then there exists

0 < f ∈ `2(V) such that (e−tLGf )t≥0 and (e−tLG′

f )t≥0

are interwoven.

▸ If G,G′ are two quantum graphs of same total length,then there exists 0 < u ∈ L2(G) such that (e−t∆Gu)t≥0

and (e−t∆G′u)t≥0 are interwoven.

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Thank you for your attention!

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Surgery of quantum graphs: eigenvalues and heat kernels · 2018-03-26 · Spectral inequalities...

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