Quantum stadium and its spectral rigidity Hong-Kun Zhang Department - - PowerPoint PPT Presentation

Quantum stadium and its spectral rigidity

Hong-Kun Zhang

Department of Mathematics and Statistics University of Massachusetts Amherst, MA, USA

July, 2019, CIRM

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Outline of the talk

1

Motivation: quantum billiards;

2

Dynamical spectral rigidity of Bunimovich stadium – joint work with Jianyu Chen, Vadim Kaloshin;

3

Main tool: Markov partition for billiards – joint work with Jianyu Chen, Fang Wang;

4

Future direction: quantum unique ergodicity.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Classical billiards are Hamiltonian systems

Billiard dynamics can be used to model Lorentz gas, Boltzmann gases and many-body particles problems in statistical mechanics, with applications in ergodic theory, and statistical physics.

modeled by a Hamiltonian function:

H(q; p) = p2

2

+ V (q) where q = (x, y) and (px, py) are the corresponding positions and momenta

• f the atom, and V (q) is a potential

vanishes inside the billiard region Q and equals to infinity outside: V (q) = 0, q ∈ Q ∞, q ∈ Qc This form of the potential guarantees a specular reflection on the boundary.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Classical billiard

The dynamics of the billiard is completely determined by the shape of its table Q. Φt : TQ → TQ is the billiard flow, defined on the tangent space

• f the table Q, preserving the Lebesgure measure.

Collision space M = ∂Q × [−π/2, π/2] = {(r, ϕ)} Billiard map F : M → M Invariant measure dµ = (2|∂Q|)−1 cos ϕdrdϕ. x Fx Q n ϕ r0 r

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Classical billiard

The dynamics of the billiard is completely determined by the shape of its table Q. Φt : TQ → TQ is the billiard flow, defined on the tangent space

• f the table Q, preserving the Lebesgure measure.

Collision space M = ∂Q × [−π/2, π/2] = {(r, ϕ)} Billiard map F : M → M Invariant measure dµ = (2|∂Q|)−1 cos ϕdrdϕ. x Fx Q n ϕ r0 r

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Quantum billiards

By the uncertainty principle in quantum mechanics, the concept

• f a trajectory becomes undefined. Instead, we say a quantum

billiard evolves according to certain quantum states. The time evolution of the quantum state of a physical system is governed by the Schrodinger equation. i¯ h∂ψ(q, t) ∂t = H¯

hψ(q, t)

is the Plank constant. H¯

h is the quantized Hamiltonian:

H¯

hψ(q, t) := − 2

2m∆ψ(q, t) + V (q)ψ(q, t) where ∇2 is the Laplacian; ψ(q, t) = Uψ(q, 0), where U = e−iH¯

h/¯

h is the unit operator.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Quantum billiards

By the uncertainty principle in quantum mechanics, the concept

• f a trajectory becomes undefined. Instead, we say a quantum

billiard evolves according to certain quantum states. The time evolution of the quantum state of a physical system is governed by the Schrodinger equation. i¯ h∂ψ(q, t) ∂t = H¯

hψ(q, t)

is the Plank constant. H¯

h is the quantized Hamiltonian:

H¯

hψ(q, t) := − 2

2m∆ψ(q, t) + V (q)ψ(q, t) where ∇2 is the Laplacian; ψ(q, t) = Uψ(q, 0), where U = e−iH¯

h/¯

h is the unit operator.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Evolution of a wave function ψ(q, t) in quantum stadium

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Stationary waves –quantum states

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Quantum states

For quantum billiards, it is enough to consider the stationary Schrodinger equation, with ψ(q, t) = ψ(q) – quantum states. Definition The stationary wave function ψ(q) is called a quantum state, if ψ(q, t) = e−itE/¯

hψ(q) solves the Schrodinger equation, or

equivalently, solve the Helmholtz equation − 2 2m∇2ψ(q) = Eψ(q) The main question in quantum chaos is to study the discreet spectrum and eigenfunctions of the operator H¯

h.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Spectrum of quantized Hamiltonian

Consider he Helmholtz equation − 2 2m∇2ψ(q) = Eψ(q) On the Hilbert space L2(Q, m), Q compact and connected, the Laplacian is selfadjoint and has only discrete spectrum: − 2 2m∇2ψn(q) = Enψn(q) The potential function imposes the Dirichlet condition: ψn(q) = 0, q ∈ ∂Q Let λn = 1

2 2mEn.

This induced to the Dirichlet-Laplacian problem: ∇2ψn(q) = −λnψ(q), q ∈ Q, ψn|∂Q = 0 √λn refers to the frequency of the wave function ψn.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Spectrum of quantized Hamiltonian

Consider he Helmholtz equation − 2 2m∇2ψ(q) = Eψ(q) On the Hilbert space L2(Q, m), Q compact and connected, the Laplacian is selfadjoint and has only discrete spectrum: − 2 2m∇2ψn(q) = Enψn(q) The potential function imposes the Dirichlet condition: ψn(q) = 0, q ∈ ∂Q Let λn = 1

2 2mEn.

This induced to the Dirichlet-Laplacian problem: ∇2ψn(q) = −λnψ(q), q ∈ Q, ψn|∂Q = 0 √λn refers to the frequency of the wave function ψn.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Spectrum of quantized Hamiltonian

Consider he Helmholtz equation − 2 2m∇2ψ(q) = Eψ(q) On the Hilbert space L2(Q, m), Q compact and connected, the Laplacian is selfadjoint and has only discrete spectrum: − 2 2m∇2ψn(q) = Enψn(q) The potential function imposes the Dirichlet condition: ψn(q) = 0, q ∈ ∂Q Let λn = 1

2 2mEn.

This induced to the Dirichlet-Laplacian problem: ∇2ψn(q) = −λnψ(q), q ∈ Q, ψn|∂Q = 0 √λn refers to the frequency of the wave function ψn.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Dirichlet-Laplacian spectrum

Q is compact and connected ⇒ H¯

h has only discrete spectrum.

It is enough to study Dirichlet-Laplacian problem: ∇2ψn(q) = −λnψ(q), q ∈ Q, ψn|∂Q = 0 The Dirichlet-Laplacian spectrum is denoted as ∆(Q), which contains all e-values 0 < λ1 ≤ λ2 ≤ · · · ≤ λn → ∞ as n → ∞, with the eigenvalues repeated according to their multiplicity. The shape of the domain Q determines the Laplacian spectrum ∆(Q). How about uniqueness? i.e. if ∆(Q) = ∆(Q′), is it true that Q = Q′ up to isometry (rotation or shift)? Definition If ∆(Q1) = ∆(Q0), then we say Q1 and Q0 are isospectral.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Dirichlet-Laplacian spectrum

Q is compact and connected ⇒ H¯

h has only discrete spectrum.

It is enough to study Dirichlet-Laplacian problem: ∇2ψn(q) = −λnψ(q), q ∈ Q, ψn|∂Q = 0 The Dirichlet-Laplacian spectrum is denoted as ∆(Q), which contains all e-values 0 < λ1 ≤ λ2 ≤ · · · ≤ λn → ∞ as n → ∞, with the eigenvalues repeated according to their multiplicity. The shape of the domain Q determines the Laplacian spectrum ∆(Q). How about uniqueness? i.e. if ∆(Q) = ∆(Q′), is it true that Q = Q′ up to isometry (rotation or shift)? Definition If ∆(Q1) = ∆(Q0), then we say Q1 and Q0 are isospectral.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Dirichlet-Laplacian spectrum

Q is compact and connected ⇒ H¯

h has only discrete spectrum.

It is enough to study Dirichlet-Laplacian problem: ∇2ψn(q) = −λnψ(q), q ∈ Q, ψn|∂Q = 0 The Dirichlet-Laplacian spectrum is denoted as ∆(Q), which contains all e-values 0 < λ1 ≤ λ2 ≤ · · · ≤ λn → ∞ as n → ∞, with the eigenvalues repeated according to their multiplicity. The shape of the domain Q determines the Laplacian spectrum ∆(Q). How about uniqueness? i.e. if ∆(Q) = ∆(Q′), is it true that Q = Q′ up to isometry (rotation or shift)? Definition If ∆(Q1) = ∆(Q0), then we say Q1 and Q0 are isospectral.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Dirichlet-Laplacian spectrum

Q is compact and connected ⇒ H¯

h has only discrete spectrum.

It is enough to study Dirichlet-Laplacian problem: ∇2ψn(q) = −λnψ(q), q ∈ Q, ψn|∂Q = 0 The Dirichlet-Laplacian spectrum is denoted as ∆(Q), which contains all e-values 0 < λ1 ≤ λ2 ≤ · · · ≤ λn → ∞ as n → ∞, with the eigenvalues repeated according to their multiplicity. The shape of the domain Q determines the Laplacian spectrum ∆(Q). How about uniqueness? i.e. if ∆(Q) = ∆(Q′), is it true that Q = Q′ up to isometry (rotation or shift)? Definition If ∆(Q1) = ∆(Q0), then we say Q1 and Q0 are isospectral.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Can ∆(Q) determines the shape of billiard table?

Weyl’s Law (1911): λn ∼ 4π area(Q) · n

• M. Kac (1966) "Can you hear the shape of a drum?"

e

• n λnt ∼ area(Q)

4πt − |∂Q| 8√πt + 1 − h 6 where h is the number of holes in Q. This is the spectral function conjecture by Kac. No – The answer is negative and was given first by J. Milnor in 1968.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Can ∆(Q) determines the shape of billiard table?

Weyl’s Law (1911): λn ∼ 4π area(Q) · n

• M. Kac (1966) "Can you hear the shape of a drum?"

e

• n λnt ∼ area(Q)

4πt − |∂Q| 8√πt + 1 − h 6 where h is the number of holes in Q. This is the spectral function conjecture by Kac. No – The answer is negative and was given first by J. Milnor in 1968.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Can one hear the shape of billiard table?

No – the first planar example was given by Gordon- Webb- Wolpert, 1992: Question: What if we take a C r (or analytic) family of isospectral domains {Qt : t ≥ 0}, obtained by deformation from Q0?

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Can one hear the shape of billiard table?

No – the first planar example was given by Gordon- Webb- Wolpert, 1992: Question: What if we take a C r (or analytic) family of isospectral domains {Qt : t ≥ 0}, obtained by deformation from Q0?

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Spectral rigidity

Definition If any C r (or analytic) family of isospectral tables {Qt} of Q0 are

• isometry. We say Q0 is C r (or analytic) spectral rigid (SR), for r > 0.

i.e. ∆(Qt) = ∆(Q0) ⇒ Qt = Q0 Conjecture (Sarnak 1990) Any planar billiard is spectral rigid (SR). However, it is rather difficult to compute or capture information from Laplacian spectrum ∆(Q)...... On the other hand, recent discoveries of the Selberg’s trace formula, Ruelle-Zeta function, etc. establish relation between the Laplacian spectrum and closed orbits (periodic orbit). Provides an alternative approach – length spectrum L(Q).

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Spectral rigidity

Definition If any C r (or analytic) family of isospectral tables {Qt} of Q0 are

• isometry. We say Q0 is C r (or analytic) spectral rigid (SR), for r > 0.

i.e. ∆(Qt) = ∆(Q0) ⇒ Qt = Q0 Conjecture (Sarnak 1990) Any planar billiard is spectral rigid (SR). However, it is rather difficult to compute or capture information from Laplacian spectrum ∆(Q)...... On the other hand, recent discoveries of the Selberg’s trace formula, Ruelle-Zeta function, etc. establish relation between the Laplacian spectrum and closed orbits (periodic orbit). Provides an alternative approach – length spectrum L(Q).

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Spectral rigidity

Definition If any C r (or analytic) family of isospectral tables {Qt} of Q0 are

• isometry. We say Q0 is C r (or analytic) spectral rigid (SR), for r > 0.

i.e. ∆(Qt) = ∆(Q0) ⇒ Qt = Q0 Conjecture (Sarnak 1990) Any planar billiard is spectral rigid (SR). However, it is rather difficult to compute or capture information from Laplacian spectrum ∆(Q)...... On the other hand, recent discoveries of the Selberg’s trace formula, Ruelle-Zeta function, etc. establish relation between the Laplacian spectrum and closed orbits (periodic orbit). Provides an alternative approach – length spectrum L(Q).

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Length spectrum L(Q)

Definition Length spectrum is the collection of length of all periodic orbits, counted by multiplicity: L(Q) := {L(γp), p ≥ 1} ∪ {m|∂Q|, m ≥ 1} Definition For any p/q ∈ Q ∩ (0, 1/2] in lowest terms, We denote a periodic

• rbit γ ∈ Γp/q is periodic with rotation number p/q.

Lmax(p/q) = max{L(γ)| γ ∈ Γp/q}. (1) The collection LM(Q) = {Lmax(p/q) p, q ∈ N} is called the marked length spectrum of Q. Lemma ((Chazarian, Melrose, Guillemin, Duistermaat, Rullel...)) The Laplace spectrum ∆(Q) determines the length spectrum L(Q).

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Results about Dynamically spectral rigidity

Definition We say a billiard on Q is Dynamically Spectral Rigid (DSR) if any smooth isospectral deformation are isometry. (Croke-Sharafutdinov 1998) Compact Rimannian manifold with negative curvature is DSR under C r deformation. (De Simoi-Kaloshin-Wei,2016) A axis-symmetric smooth convex billiard near the circle is DSR. (Callis-Kaloshin-Sorrentino 2019) A C r generic axis-symmetric smooth convex billiard is DSR. More exactly, there is a C r open and dense set of axis-symmetric smooth convex billiard. No results yet for billiards with piecewise boundary

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Theorem (Jianyu Chen- Vadim Kaloshin- H.Zh (2019)) Bunimovich stadium is DSR under piecewise analytic deformation. In addition, Bunimovich squash-like stadium is DSR under piecewise analytic deformation.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Theorem Let τ∗ = |γ∗| be the length of the periodic-2 orbit between two curved boundaries, then the limits exist: −B1/2 := lim

q→∞

• Lmax

q − 1 2q

• − 2qτ ∗
• ,

(2) − log λ1/2 := lim

q→∞

1 q log

• Lmax

q − 1 2q

• − 2qτ ∗ + B1/2
• , (3)

C1/2 := lim

q→∞

• λ1/2

q

• Lmax

Ω

q − 1 2q

• − 2qτ ∗ + B1/2
• ,(4)

where q is an odd integer. In the Aubry-Mather theory, B1/2 is referred to the Peierls’ Barrier function evaluated on a certain homoclinic orbit of γ∗, and λ1/2 is the eigenvalue of the linearization of the billiard map along γ∗.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Theorem Let τ∗ = |γ∗|, then C1/2 = lim

q→∞

• λ1/2

q

• Lmax

Ω

q − 1 2q

• − 2qτ ∗ + B1/2
• ,

(5) Moreover K1 and K2 can be computed: λ 1

2 + λ−1 1 2

= 4 (τ ∗K1 − 1) (τ ∗K2 − 1) − 2, (6)

• C 1

1 2 − C 2 1 2

λ 1

2 + 1

2

• C 1

1 2 + C 2 1 2

λ 1

2 − 1

2 =

• 1 −

1 τ ∗K1 2 −

• 1 −

1 τ ∗K2 2

• 1 −

1 τ ∗K1 − 1 τ ∗K2 2 + 3 . (7)

Hong-Kun Zhang Quantum stadium and its spectral rigidity

DSR under C r deformation

Let Q0 be a Bunimovich stadium or squash stadium, with fixed flat sides of length 1. Let {Qt} be any family of billiards obtained by C r deformation

• f the two convex curves of Q0.

Theorem (Jianyu Chen-Vadim Kaloshin-H.Zhang (in progress)) Let {Qt} be any family of C r isospectral deformation of Q0. Then Qt and Q0 are isometry. i.e. Bunimovich (squash) stadium is DSR under C r deformation.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Construction of Markov partition for hyperbolic billiards

Theorem (Jianyu Chen,Fang Wang&H.Zh (2019)) The billiard system (F, M) has a countable Markov partition: M =

∞

• n=1

n−1

• k=0

F kR∗

n.

(1) For each n ≥ 1, k = 0, · · · , n − 1, the Markov element F kR∗

n is a

fat cantor set; (2) all top elements are disjoint (except the base), n, m ≥ 1, k = 1, · · · , n − 1, l = 1, · · · , m − 1 µ(F kR∗

n ∩ F lR∗ m) = 0

if (n, k) = (m, l). (3) F nR∗

n return to all R∗ m, m ≥ 1, markovly (along unstable

direction).

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Construction of Markov partition for hyperbolic billiards

Theorem (Jianyu Chen,Fang Wang&H.Zh (2019)) The billiard system (F, M) has a countable Markov partition: M =

∞

• n=1

n−1

• k=0

F kR∗

n.

(1) For each n ≥ 1, k = 0, · · · , n − 1, the Markov element F kR∗

n is a

fat cantor set; (2) all top elements are disjoint (except the base), n, m ≥ 1, k = 1, · · · , n − 1, l = 1, · · · , m − 1 µ(F kR∗

n ∩ F lR∗ m) = 0

if (n, k) = (m, l). (3) F nR∗

n return to all R∗ m, m ≥ 1, markovly (along unstable

direction).

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Construction of Markov partition for hyperbolic billiards

Theorem (Jianyu Chen,Fang Wang&H.Zh (2019)) The billiard system (F, M) has a countable Markov partition: M =

∞

• n=1

n−1

• k=0

F kR∗

n.

(1) For each n ≥ 1, k = 0, · · · , n − 1, the Markov element F kR∗

n is a

fat cantor set; (2) all top elements are disjoint (except the base), n, m ≥ 1, k = 1, · · · , n − 1, l = 1, · · · , m − 1 µ(F kR∗

n ∩ F lR∗ m) = 0

if (n, k) = (m, l). (3) F nR∗

n return to all R∗ m, m ≥ 1, markovly (along unstable

direction).

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Construction of hyperbolic set

U W s(x) W u

δ (x)

Γs Γu

Figure: R∗ and U

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Theorem (Chen,Wang&Zh (2019) – Markov hyperbolic set) There is a hyperbolic product set R∗ ⊂ M of positive µ-measure such that the following properties hold: (1) Decomposition of R∗: there is a countable family of closed s-subsets R∗

n ⊂ R∗ such that R∗ = ∪n≥1R∗ n (mod µ).

(2) First return is Markov: for any R∗

n with µ(R∗ n) > 0, F nR∗ n

returns to R∗ for the first time:

(i) F nR∗

n is a u-subset of R∗;

(ii) F iR∗

n ∩ R∗ = ∅ (mod µ) for any i = 1, 2, · · · , n − 1.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Theorem (Chen,Wang&Zh (2019) – Markov hyperbolic set) There is a hyperbolic product set R∗ ⊂ M of positive µ-measure such that the following properties hold: (1) Decomposition of R∗: there is a countable family of closed s-subsets R∗

n ⊂ R∗ such that R∗ = ∪n≥1R∗ n (mod µ).

(2) First return is Markov: for any R∗

n with µ(R∗ n) > 0, F nR∗ n

returns to R∗ for the first time:

(i) F nR∗

n is a u-subset of R∗;

(ii) F iR∗

n ∩ R∗ = ∅ (mod µ) for any i = 1, 2, · · · , n − 1.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Rm Rn F mRm FRm F 2Rm F 3Rm F nRn M

Hong-Kun Zhang Quantum stadium and its spectral rigidity

R∗ = ∆0 ∆1 ∆2 R1 R2 FR2 Rn FRn FR1 M

Hong-Kun Zhang Quantum stadium and its spectral rigidity

¯ ∆0 ¯ ∆1 ¯ ∆2 ¯ ∆0,1 ¯ ∆0,n ¯ F ¯ ∆0,n ¯ ∆

Hong-Kun Zhang Quantum stadium and its spectral rigidity

A new Markov twoer tower for billiards

non-proper returns

¯ ∆0 ¯ ∆1 ¯ ∆2 ¯ ∆0,1 ¯ ∆0,n ¯ ∆0,n ¯ ∆0,n ¯ F ¯ ∆0,n ¯ F ¯ ∆0,n ¯ ∆

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Improved Young tower for billiards

non-proper returns

R∗ = ∆0 ∆1 ∆2 R1 R2 FR2 Rn Rn FRn FRn FR1 M

Remark: The Young tower constructed by Young 1998 for Sinai billiards does not establish a conjugacy between the billiard map and Gibbs-Markov map.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Conjugate to Gibbs Markov maps

Let F0 be the σ-algebra generated by the countable Markov partition for (F, M), and ξ ∈ F0 a random variable that takes distinct value on each component of the partition. F0 ⊂ F1 ⊂ · · · ⊂ Fn · · · is a filtration with Fn = σ(ξ, ξ ◦ F, · · · , ξ ◦ F n). For any Holder observable f on M, let ¯ f = E(f |F0). Then Xn := ¯ f ◦ F n defines a Markov process, satisfying various limiting theorems. |f ◦ F n − ¯ f ◦ F n| = |(f − ¯ f )|Fn); Thus we are able to extend most existing limiting theorems for Gibbs Markov map to hyperbolic systems with singularities.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Future directions: Conjecture of QUE

Spectrum properties of Quantized Hamiltonian with Dirichlet boundary condition: − 2 2m∇2ψn(q) = Enψn(q) ψn(q) = 0, q ∈ ∂Q The quantum states {ψn} forms an ONB in L2(Q, m), with < ψm, ψn >= δm,n; ψn are C ∞ functions. Probability measures µψn: dµψn(x) = |ψn|2(x)dx What can we say about the accumulating points of {µψn}?

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Quantum Ergodicity

Definition Any limiting points in {µψn} is called a semi-classical measure, denoted as µsc ∈ Msc. Any µsc is invariant under the billiard flow. i.e. Msc ⊂ Minv. We say the billiard is quantum ergodic if m ∈ Msc, i.e. there exists a dense-one subset S ⊂ N, such that lim

n→∞,n∈S µψn = m

S ⊂ N is a dense one set is lim

N→∞

#{j ∈ S : j < N} N = 1 Theorem (Schnirelman (1974), Zelditch(1989) Gerard, Leichtman,(1993)): If the classical billiard Φt is ergodic, then it is quantum ergodic.

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Quantum unique ergodicity

Quantum ergodicty implies that most eigenfunctions of {ψn} become equidistributed on Q in the semiclassical limit. Definition We say the billiard is Quantum Unique Ergodic (QUE) if Msc = {m}, i.e. lim

n→∞ µψn = m

Conjecture (Sarnak): Ergodic billiards are Quantum Unique Ergodic. Definition Let limk→∞ µψnk = µsc. We say that the sequence of eigenfunctions ψnk exhibits strong scarring if µsc is a measure that is entirely supported on periodic classical trajectories. We say that the sequence

• f eigenfunctions ψnk exhibits partial scarring on a periodic classical

trajectory γp if µsc(γp) > 0. Theorem ( Anantharaman and Nonnenmacher 2008) âŁœScarring⣞

Hong-Kun Zhang Quantum stadium and its spectral rigidity

Quantum scars for Stadium

Question: Maybe Bunimovich stadium is not QUE?

Hong-Kun Zhang Quantum stadium and its spectral rigidity

(Andrew Hassell 2018): Bunimovich stadium is NOT QUE. Question: Is Sinai billiard QUE? – open; Question: Are there any other ergodic billiards that are not QUE?

Hong-Kun Zhang Quantum stadium and its spectral rigidity