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Quantum graphs where back-scattering is prohibited Brian Winn School of Mathematics Loughborough University 26th February 2008 Brian Winn Quantum graphs without back-scattering Credits Joint work with Jon Harrison and Uzy Smilansky

SLIDE 1

Quantum graphs where back-scattering is prohibited

Brian Winn

School of Mathematics Loughborough University

26th February 2008

Brian Winn Quantum graphs without back-scattering

SLIDE 2

Credits

Joint work with Jon Harrison and Uzy Smilansky Journal of Physics A 40 14181–14193.

Brian Winn Quantum graphs without back-scattering

SLIDE 3

A puzzle

Find an n × n unitary matrix σ: with diagonal entries 0, and

ff-diagonal entries with absolute value (n − 1)−1/2.

For example (2 × 2) σ = 1 1

Hint: n = 3 is impossible. . .

Brian Winn Quantum graphs without back-scattering

SLIDE 4

A puzzle

Find an n × n unitary matrix σ: with diagonal entries 0, and

ff-diagonal entries with absolute value (n − 1)−1/2.

For example (2 × 2) σ = 1 1

Hint: n = 3 is impossible. . .

Brian Winn Quantum graphs without back-scattering

SLIDE 5

A puzzle

Find an n × n unitary matrix σ: with diagonal entries 0, and

ff-diagonal entries with absolute value (n − 1)−1/2.

For example (2 × 2) σ = 1 1

Hint: n = 3 is impossible. . .

Brian Winn Quantum graphs without back-scattering

SLIDE 6

A puzzle

Find an n × n unitary matrix σ: with diagonal entries 0, and

ff-diagonal entries with absolute value (n − 1)−1/2.

For example (2 × 2) σ = 1 1

Hint: n = 3 is impossible. . .

Brian Winn Quantum graphs without back-scattering

SLIDE 7

A puzzle

Find an n × n unitary matrix σ: with diagonal entries 0, and

ff-diagonal entries with absolute value (n − 1)−1/2.

For example (2 × 2) σ = 1 1

Hint: n = 3 is impossible. . .

Brian Winn Quantum graphs without back-scattering

SLIDE 8

What is a quantum graph?

A metric graph has bonds that have lengths L1, . . . , Lv > 0. Standing waves satisfy −d2ψj dx2 = k2ψj + Boundary conditions j = 1, . . . , v. For k = k0, k1, k2, . . . the spectrum of the quantum graph.

Brian Winn Quantum graphs without back-scattering

SLIDE 9

What is a quantum graph?

A metric graph has bonds that have lengths L1, . . . , Lv > 0. Standing waves satisfy −d2ψj dx2 = k2ψj + Boundary conditions j = 1, . . . , v. For k = k0, k1, k2, . . . the spectrum of the quantum graph.

Brian Winn Quantum graphs without back-scattering

SLIDE 10

What is a quantum graph?

A metric graph has bonds that have lengths L1, . . . , Lv > 0. Standing waves satisfy −d2ψj dx2 = k2ψj + Boundary conditions j = 1, . . . , v. For k = k0, k1, k2, . . . the spectrum of the quantum graph.

Brian Winn Quantum graphs without back-scattering

SLIDE 11

Scattering at a vertex

(Boundary conditions for the differential equation)

An incoming wave is scattered at a vertex 1 3 2 eikx Scattering is controlled by a d × d unitary matrix σ. d is the degree of the vertex. We do not say anything about the process causing the scattering.

Brian Winn Quantum graphs without back-scattering

SLIDE 12

Scattering at a vertex

(Boundary conditions for the differential equation)

An incoming wave is scattered at a vertex 1 3 2 eikx σ11eikx σ12eikx σ13eikx Scattering is controlled by a d × d unitary matrix σ. d is the degree of the vertex. We do not say anything about the process causing the scattering.

Brian Winn Quantum graphs without back-scattering

SLIDE 13

Scattering at a vertex

(Boundary conditions for the differential equation)

An incoming wave is scattered at a vertex 1 3 2 eikx σ11eikx σ12eikx σ13eikx Scattering is controlled by a d × d unitary matrix σ. d is the degree of the vertex. We do not say anything about the process causing the scattering.

Brian Winn Quantum graphs without back-scattering

SLIDE 14

Scattering at a vertex

(Boundary conditions for the differential equation)

An incoming wave is scattered at a vertex 1 3 2 eikx σ11eikx σ12eikx σ13eikx Scattering is controlled by a d × d unitary matrix σ. d is the degree of the vertex. We do not say anything about the process causing the scattering.

Brian Winn Quantum graphs without back-scattering

SLIDE 15

Scattering at a vertex

(Boundary conditions for the differential equation)

An incoming wave is scattered at a vertex 1 3 2 eikx σ11eikx σ12eikx σ13eikx Scattering is controlled by a d × d unitary matrix σ. d is the degree of the vertex. We do not say anything about the process causing the scattering.

Brian Winn Quantum graphs without back-scattering

SLIDE 16

The quantum evolution operator

Collect all entries of vertex scattering matrices σ in a 2v × 2v matrix S. Indexing is by directed bonds. In this example S =         σ12 σ13 σ11 1 1 1 σ22 σ23 σ21 σ32 σ33 σ31         .

Brian Winn Quantum graphs without back-scattering

SLIDE 17

The quantum evolution operator

Collect all entries of vertex scattering matrices σ in a 2v × 2v matrix S. Indexing is by directed bonds. In this example S =         σ12 σ13 σ11 1 1 1 σ22 σ23 σ21 σ32 σ33 σ31         .

Brian Winn Quantum graphs without back-scattering

SLIDE 18

The quantum evolution operator

Collect all entries of vertex scattering matrices σ in a 2v × 2v matrix S. Indexing is by directed bonds. In this example S =         σ12 σ13 σ11 1 1 1 σ22 σ23 σ21 σ32 σ33 σ31         .

Brian Winn Quantum graphs without back-scattering

SLIDE 19

The quantum evolution operator

Collect all entries of vertex scattering matrices σ in a 2v × 2v matrix S. Indexing is by directed bonds. 1 2 3 4 5 6 In this example S =         σ12 σ13 σ11 1 1 1 σ22 σ23 σ21 σ32 σ33 σ31         .

Brian Winn Quantum graphs without back-scattering

SLIDE 20

The quantum evolution operator

Collect all entries of vertex scattering matrices σ in a 2v × 2v matrix S. Indexing is by directed bonds. Scattering matrix at centre σ =   σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33  . 1 2 3 4 5 6 In this example S =         σ12 σ13 σ11 1 1 1 σ22 σ23 σ21 σ32 σ33 σ31         .

Brian Winn Quantum graphs without back-scattering

SLIDE 21

The quantum evolution operator

Collect all entries of vertex scattering matrices σ in a 2v × 2v matrix S. Indexing is by directed bonds. Scattering matrix at centre σ =   σ11 σ12 σ13 σ21 σ22 σ23 σ31 σ32 σ33  . 1 2 3 4 5 6 In this example S =         σ12 σ13 σ11 1 1 1 σ22 σ23 σ21 σ32 σ33 σ31         .

Brian Winn Quantum graphs without back-scattering

SLIDE 22

The quantum evolution operator

Continued

Waves travelling along a bond of length L acquire a phase eikL. Put these phases into a 2v × 2v diagonal matrix D(k). Define the quantum evolution operator U(k) = D(k)S. The spectrum There is a standing wave of energy k2 iff det(I − U(k)) = 0. A sequence (kn)∞

n=1 of “eigenvalues”.

Alternatively: Use the von Neumann theory to construct self-adjoint extensions of the Laplace operator. . . (Kostrykin & Schrader approach.)

Brian Winn Quantum graphs without back-scattering

SLIDE 23

The quantum evolution operator

Continued

Waves travelling along a bond of length L acquire a phase eikL. Put these phases into a 2v × 2v diagonal matrix D(k). Define the quantum evolution operator U(k) = D(k)S. The spectrum There is a standing wave of energy k2 iff det(I − U(k)) = 0. A sequence (kn)∞

n=1 of “eigenvalues”.

Alternatively: Use the von Neumann theory to construct self-adjoint extensions of the Laplace operator. . . (Kostrykin & Schrader approach.)

Brian Winn Quantum graphs without back-scattering

SLIDE 24

The quantum evolution operator

Continued

Waves travelling along a bond of length L acquire a phase eikL. Put these phases into a 2v × 2v diagonal matrix D(k). Define the quantum evolution operator U(k) = D(k)S. The spectrum There is a standing wave of energy k2 iff det(I − U(k)) = 0. A sequence (kn)∞

n=1 of “eigenvalues”.

Alternatively: Use the von Neumann theory to construct self-adjoint extensions of the Laplace operator. . . (Kostrykin & Schrader approach.)

Brian Winn Quantum graphs without back-scattering

SLIDE 25

The quantum evolution operator

Continued

Waves travelling along a bond of length L acquire a phase eikL. Put these phases into a 2v × 2v diagonal matrix D(k). Define the quantum evolution operator U(k) = D(k)S. The spectrum There is a standing wave of energy k2 iff det(I − U(k)) = 0. A sequence (kn)∞

n=1 of “eigenvalues”.

Alternatively: Use the von Neumann theory to construct self-adjoint extensions of the Laplace operator. . . (Kostrykin & Schrader approach.)

Brian Winn Quantum graphs without back-scattering

SLIDE 26

The quantum evolution operator

Continued

Waves travelling along a bond of length L acquire a phase eikL. Put these phases into a 2v × 2v diagonal matrix D(k). Define the quantum evolution operator U(k) = D(k)S. The spectrum There is a standing wave of energy k2 iff det(I − U(k)) = 0. A sequence (kn)∞

n=1 of “eigenvalues”.

Alternatively: Use the von Neumann theory to construct self-adjoint extensions of the Laplace operator. . . (Kostrykin & Schrader approach.)

Brian Winn Quantum graphs without back-scattering

SLIDE 27

A classical evolution operator

Classical dynamics is a Markov process on the directed bonds, with matrix of transition probabilities M, where Mbb′ = |Ubb′|2. Since U is unitary, M is stochastic (indeed doubly stochastic). M has an eigenvalue 1 and all other eigenvalues have absolute value 1.

Brian Winn Quantum graphs without back-scattering

SLIDE 28

A classical evolution operator

Classical dynamics is a Markov process on the directed bonds, with matrix of transition probabilities M, where Mbb′ = |Ubb′|2. Since U is unitary, M is stochastic (indeed doubly stochastic). M has an eigenvalue 1 and all other eigenvalues have absolute value 1.

Brian Winn Quantum graphs without back-scattering

SLIDE 29

A classical evolution operator

Classical dynamics is a Markov process on the directed bonds, with matrix of transition probabilities M, where Mbb′ = |Ubb′|2. Since U is unitary, M is stochastic (indeed doubly stochastic). M has an eigenvalue 1 and all other eigenvalues have absolute value 1.

Brian Winn Quantum graphs without back-scattering

SLIDE 30

A classical evolution operator

Classical dynamics is a Markov process on the directed bonds, with matrix of transition probabilities M, where Mbb′ = |Ubb′|2. Since U is unitary, M is stochastic (indeed doubly stochastic). M has an eigenvalue 1 and all other eigenvalues have absolute value 1. 1 ∆

Brian Winn Quantum graphs without back-scattering

SLIDE 31

A classical evolution operator

Classical dynamics is a Markov process on the directed bonds, with matrix of transition probabilities M, where Mbb′ = |Ubb′|2. Since U is unitary, M is stochastic (indeed doubly stochastic). M has an eigenvalue 1 and all other eigenvalues have absolute value 1. 1 ∆ Denote by ∆ the spectral gap.

Brian Winn Quantum graphs without back-scattering

SLIDE 32

Random matrix conjecture

Conjecture (Tanner) For a sequence of quantum graphs with v → ∞, the statistics of eigenvalues converge to random matrix theory if v∆ → ∞. Gnutzmann & Altland approach. True if √v∆ → ∞. (Not a periodic orbit theory). Are there “nicer” choices of vertex scattering matrix? Quantum ergodicity? Scarring??

Brian Winn Quantum graphs without back-scattering

SLIDE 33

Random matrix conjecture

Conjecture (Tanner) For a sequence of quantum graphs with v → ∞, the statistics of eigenvalues converge to random matrix theory if v∆ → ∞. Gnutzmann & Altland approach. True if √v∆ → ∞. (Not a periodic orbit theory). Are there “nicer” choices of vertex scattering matrix? Quantum ergodicity? Scarring??

Brian Winn Quantum graphs without back-scattering

SLIDE 34

Random matrix conjecture

Conjecture (Tanner) For a sequence of quantum graphs with v → ∞, the statistics of eigenvalues converge to random matrix theory if v∆ → ∞. Gnutzmann & Altland approach. True if √v∆ → ∞. (Not a periodic orbit theory). Are there “nicer” choices of vertex scattering matrix? Quantum ergodicity? Scarring??

Brian Winn Quantum graphs without back-scattering

SLIDE 35

Random matrix conjecture

Conjecture (Tanner) For a sequence of quantum graphs with v → ∞, the statistics of eigenvalues converge to random matrix theory if v∆ → ∞. Gnutzmann & Altland approach. True if √v∆ → ∞. (Not a periodic orbit theory). Are there “nicer” choices of vertex scattering matrix? Quantum ergodicity? Scarring??

Brian Winn Quantum graphs without back-scattering

SLIDE 36

Random matrix conjecture

Conjecture (Tanner) For a sequence of quantum graphs with v → ∞, the statistics of eigenvalues converge to random matrix theory if v∆ → ∞. Gnutzmann & Altland approach. True if √v∆ → ∞. (Not a periodic orbit theory). Are there “nicer” choices of vertex scattering matrix? Quantum ergodicity? Scarring??

Brian Winn Quantum graphs without back-scattering

SLIDE 37

Possible vertex scattering matrices

Neumann σ[N]

jk = 2

d − δjk. Back-scattering strongly favoured. Fourier transform σ[F]

jk =

1 √ d e−2πijk/d. All amplitudes equal.

Brian Winn Quantum graphs without back-scattering

SLIDE 38

Possible vertex scattering matrices

Neumann σ[N]

jk = 2

d − δjk. Back-scattering strongly favoured. Fourier transform σ[F]

jk =

1 √ d e−2πijk/d. All amplitudes equal.

Brian Winn Quantum graphs without back-scattering

SLIDE 39

Possible vertex scattering matrices

Neumann σ[N]

jk = 2

d − δjk. Back-scattering strongly favoured. Fourier transform σ[F]

jk =

1 √ d e−2πijk/d. All amplitudes equal. Equi-transmitting |σjk|2 = 1 − δjk d − 1 . All forward amplitudes equal; back-scattering weighted zero.

Brian Winn Quantum graphs without back-scattering

SLIDE 40

Examples

Partial solution to puzzle

d = 3, no examples (impossible). d = 4, σ = 1 √ 3     1 1 1 1 −1 1 1 1 −1 1 −1 1    . d = 5, σ = 1 2       1 1 1 1 1 1 ω ω2 1 1 ω2 ω 1 ω ω2 1 1 ω2 ω 1       , where ω = e2πi/3. Examples for d any multiple of 4, up to 184, related to existence of skew Hadamard matrices. Examples for d = p + 1, for all odd primes p. Examples for d = 2n.

Brian Winn Quantum graphs without back-scattering

SLIDE 41

Examples

Partial solution to puzzle

d = 3, no examples (impossible). d = 4, σ = 1 √ 3     1 1 1 1 −1 1 1 1 −1 1 −1 1    . d = 5, σ = 1 2       1 1 1 1 1 1 ω ω2 1 1 ω2 ω 1 ω ω2 1 1 ω2 ω 1       , where ω = e2πi/3. Examples for d any multiple of 4, up to 184, related to existence of skew Hadamard matrices. Examples for d = p + 1, for all odd primes p. Examples for d = 2n.

Brian Winn Quantum graphs without back-scattering

SLIDE 42

Examples

Partial solution to puzzle

d = 3, no examples (impossible). d = 4, σ = 1 √ 3     1 1 1 1 −1 1 1 1 −1 1 −1 1    . d = 5, σ = 1 2       1 1 1 1 1 1 ω ω2 1 1 ω2 ω 1 ω ω2 1 1 ω2 ω 1       , where ω = e2πi/3. Examples for d any multiple of 4, up to 184, related to existence of skew Hadamard matrices. Examples for d = p + 1, for all odd primes p. Examples for d = 2n.

Brian Winn Quantum graphs without back-scattering

SLIDE 43

Examples

Partial solution to puzzle

d = 3, no examples (impossible). d = 4, σ = 1 √ 3     1 1 1 1 −1 1 1 1 −1 1 −1 1    . d = 5, σ = 1 2       1 1 1 1 1 1 ω ω2 1 1 ω2 ω 1 ω ω2 1 1 ω2 ω 1       , where ω = e2πi/3. Examples for d any multiple of 4, up to 184, related to existence of skew Hadamard matrices. Examples for d = p + 1, for all odd primes p. Examples for d = 2n.

Brian Winn Quantum graphs without back-scattering

SLIDE 44

Examples

Partial solution to puzzle

d = 3, no examples (impossible). d = 4, σ = 1 √ 3     1 1 1 1 −1 1 1 1 −1 1 −1 1    . d = 5, σ = 1 2       1 1 1 1 1 1 ω ω2 1 1 ω2 ω 1 ω ω2 1 1 ω2 ω 1       , where ω = e2πi/3. Examples for d any multiple of 4, up to 184, related to existence of skew Hadamard matrices. Examples for d = p + 1, for all odd primes p. Examples for d = 2n.

Brian Winn Quantum graphs without back-scattering

SLIDE 45

Examples

Partial solution to puzzle

d = 3, no examples (impossible). d = 4, σ = 1 √ 3     1 1 1 1 −1 1 1 1 −1 1 −1 1    . d = 5, σ = 1 2       1 1 1 1 1 1 ω ω2 1 1 ω2 ω 1 ω ω2 1 1 ω2 ω 1       , where ω = e2πi/3. Examples for d any multiple of 4, up to 184, related to existence of skew Hadamard matrices. Examples for d = p + 1, for all odd primes p. Examples for d = 2n.

Brian Winn Quantum graphs without back-scattering

SLIDE 46

The trace formula

For suitable test functions h,

∞

h(kn) =

b Lb

π ˆ h(0) + 1 π

Apℓp rp ˆ h(ℓp), where ℓp are (metric) lengths of periodic orbits, rp is repetition number and Ap = Sb1,b2Sb2,b3 · · · Sbn,b1 is product of elements of S accumulated on the orbit. With equi-transmitting scattering matrices, back-tracking

rbits are eliminated.

c.f. Ihara-Selberg zeta function.

Brian Winn Quantum graphs without back-scattering

SLIDE 47

The trace formula

For suitable test functions h,

∞

h(kn) =

b Lb

π ˆ h(0) + 1 π

Apℓp rp ˆ h(ℓp), where ℓp are (metric) lengths of periodic orbits, rp is repetition number and Ap = Sb1,b2Sb2,b3 · · · Sbn,b1 is product of elements of S accumulated on the orbit. With equi-transmitting scattering matrices, back-tracking

rbits are eliminated.

c.f. Ihara-Selberg zeta function.

Brian Winn Quantum graphs without back-scattering

SLIDE 48

The trace formula

For suitable test functions h,

∞

h(kn) =

b Lb

π ˆ h(0) + 1 π

Apℓp rp ˆ h(ℓp), where ℓp are (metric) lengths of periodic orbits, rp is repetition number and Ap = Sb1,b2Sb2,b3 · · · Sbn,b1 is product of elements of S accumulated on the orbit. With equi-transmitting scattering matrices, back-tracking

rbits are eliminated.

c.f. Ihara-Selberg zeta function.

Brian Winn Quantum graphs without back-scattering

SLIDE 49

The trace formula

For suitable test functions h,

∞

h(kn) =

b Lb

π ˆ h(0) + 1 π

Apℓp rp ˆ h(ℓp), where ℓp are (metric) lengths of periodic orbits, rp is repetition number and Ap = Sb1,b2Sb2,b3 · · · Sbn,b1 is product of elements of S accumulated on the orbit. With equi-transmitting scattering matrices, back-tracking

rbits are eliminated.

c.f. Ihara-Selberg zeta function.

Brian Winn Quantum graphs without back-scattering

SLIDE 50

The trace formula

For suitable test functions h,

∞

h(kn) =

b Lb

π ˆ h(0) + 1 π

Apℓp rp ˆ h(ℓp), where ℓp are (metric) lengths of periodic orbits, rp is repetition number and Ap = Sb1,b2Sb2,b3 · · · Sbn,b1 is product of elements of S accumulated on the orbit. With equi-transmitting scattering matrices, back-tracking

rbits are eliminated.

c.f. Ihara-Selberg zeta function.

Brian Winn Quantum graphs without back-scattering

SLIDE 51

Further Ihara connections

In a regular graph all vertices have the same degree. Theorem 1 (Harrison, Smilansky, W.) For a regular quantum graph with equi-transmitting vertex scattering matrices the eigenvalues of M are (up to scaling) at the positions of the poles of the Ihara-Selberg zeta function. This allows us to prove: Theorem 2 (Harrison, Smilansky, W.) For a regular quantum graph with equi-transmitting vertex scattering matrices the spectral gap is strictly greater than the spectral gap for the same graph with Neumann or Fourier transform scattering matrices.

Brian Winn Quantum graphs without back-scattering

SLIDE 52

Further Ihara connections

In a regular graph all vertices have the same degree. Theorem 1 (Harrison, Smilansky, W.) For a regular quantum graph with equi-transmitting vertex scattering matrices the eigenvalues of M are (up to scaling) at the positions of the poles of the Ihara-Selberg zeta function. This allows us to prove: Theorem 2 (Harrison, Smilansky, W.) For a regular quantum graph with equi-transmitting vertex scattering matrices the spectral gap is strictly greater than the spectral gap for the same graph with Neumann or Fourier transform scattering matrices.

Brian Winn Quantum graphs without back-scattering

SLIDE 53

Further Ihara connections

In a regular graph all vertices have the same degree. Theorem 1 (Harrison, Smilansky, W.) For a regular quantum graph with equi-transmitting vertex scattering matrices the eigenvalues of M are (up to scaling) at the positions of the poles of the Ihara-Selberg zeta function. This allows us to prove: Theorem 2 (Harrison, Smilansky, W.) For a regular quantum graph with equi-transmitting vertex scattering matrices the spectral gap is strictly greater than the spectral gap for the same graph with Neumann or Fourier transform scattering matrices.

Brian Winn Quantum graphs without back-scattering

SLIDE 54

Spectral statistics for an example

We considered the 5-regular graph

15 13 8 10 7 18 14 6 1 19 4 20 12 16 3 17 9 11 2 5

Brian Winn Quantum graphs without back-scattering

SLIDE 55

Spectral statistics for an example

We considered the 5-regular graph

15 13 8 10 7 18 14 6 1 19 4 20 12 16 3 17 9 11 2 5

with vertex scattering matrix σ = 1 2       1 1 1 1 1 1 ω ω2 1 1 ω2 ω 1 ω ω2 1 1 ω2 ω 1       at each vertex.

Brian Winn Quantum graphs without back-scattering

SLIDE 56

Spectral statistics for an example

continued

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12

Quantum graph GOE GUE Quantum graph GOE GUE

P(s) s L V(L) Nearest neighbour spacing Number variance

Brian Winn Quantum graphs without back-scattering

SLIDE 57

Lies

To simplify the exposition

Specifying vertex scattering matrix is not equivalent to self-adjoint extension of the Laplace operator. I omitted a technical condition from Theorem 2. I did not compute eigenvalues to draw the figures—instead I averaged statistics of eigenphases of U(k) over bond lengths. There are no lies in the article.

Brian Winn Quantum graphs without back-scattering

SLIDE 58

Lies

To simplify the exposition

Specifying vertex scattering matrix is not equivalent to self-adjoint extension of the Laplace operator. I omitted a technical condition from Theorem 2. I did not compute eigenvalues to draw the figures—instead I averaged statistics of eigenphases of U(k) over bond lengths. There are no lies in the article.

Brian Winn Quantum graphs without back-scattering

SLIDE 59

Lies

To simplify the exposition

Specifying vertex scattering matrix is not equivalent to self-adjoint extension of the Laplace operator. I omitted a technical condition from Theorem 2. I did not compute eigenvalues to draw the figures—instead I averaged statistics of eigenphases of U(k) over bond lengths. There are no lies in the article.

Brian Winn Quantum graphs without back-scattering

SLIDE 60

Lies

To simplify the exposition

Specifying vertex scattering matrix is not equivalent to self-adjoint extension of the Laplace operator. I omitted a technical condition from Theorem 2. I did not compute eigenvalues to draw the figures—instead I averaged statistics of eigenphases of U(k) over bond lengths. There are no lies in the article.

Brian Winn Quantum graphs without back-scattering

SLIDE 61

Outlook

Equi-transmitting matrices of other dimensions. First open dimension is d = 7. Can one “hear” the shape of quantum graphs with equi-transmitting scattering matrices? Gutkin & Smilansky proof breaks down here. Are there other interesting connections to discrete graph

bjects?

Brian Winn Quantum graphs without back-scattering

SLIDE 62

Outlook

Equi-transmitting matrices of other dimensions. First open dimension is d = 7. Can one “hear” the shape of quantum graphs with equi-transmitting scattering matrices? Gutkin & Smilansky proof breaks down here. Are there other interesting connections to discrete graph

bjects?

Brian Winn Quantum graphs without back-scattering

SLIDE 63

Outlook

Equi-transmitting matrices of other dimensions. First open dimension is d = 7. Can one “hear” the shape of quantum graphs with equi-transmitting scattering matrices? Gutkin & Smilansky proof breaks down here. Are there other interesting connections to discrete graph

bjects?

Brian Winn Quantum graphs without back-scattering