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 Motivation: quantum billiards; 1 Dynamical spectral rigidity of Bunimovich stadium 2 – joint work with Jianyu Chen, Vadim Kaloshin; Main tool: Markov partition for billiards 3 – joint work with Jianyu Chen, Fang Wang; Future direction: quantum unique ergodicity. 4 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 ) = � p � 2 + V ( q ) 2 where q = ( x , y ) and ( p x , p y ) are the corresponding positions and momenta of the atom, and V ( q ) is a potential vanishes inside the billiard region Q and equals to infinity outside: � 0 , q ∈ Q V ( q ) = q ∈ Q c ∞ , 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 . Φ t : TQ → TQ is the billiard flow, defined on the tangent space of the table Q , preserving the Lebesgure measure. Collision space M = ∂ Q × [ − π/ 2 , π/ 2 ] = { ( r , ϕ ) } Billiard map F : M → M Invariant measure d µ = ( 2 | ∂ Q | ) − 1 cos ϕ drd ϕ . n Q ϕ Fx x r 0 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 . Φ t : TQ → TQ is the billiard flow, defined on the tangent space of the table Q , preserving the Lebesgure measure. Collision space M = ∂ Q × [ − π/ 2 , π/ 2 ] = { ( r , ϕ ) } Billiard map F : M → M Invariant measure d µ = ( 2 | ∂ Q | ) − 1 cos ϕ drd ϕ . n Q ϕ Fx x r 0 r Hong-Kun Zhang Quantum stadium and its spectral rigidity

Quantum billiards By the uncertainty principle in quantum mechanics, the concept of 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. h ∂ψ ( q , t ) i ¯ = H ¯ h ψ ( q , t ) ∂ t � is the Plank constant. H ¯ h is the quantized Hamiltonian: h ψ ( q , t ) := − � 2 H ¯ 2 m ∆ ψ ( q , t ) + V ( q ) ψ ( q , t ) where ∇ 2 is the Laplacian; h / ¯ h is the unit operator. ψ ( q , t ) = U ψ ( q , 0 ) , where U = e − iH ¯ Hong-Kun Zhang Quantum stadium and its spectral rigidity

Quantum billiards By the uncertainty principle in quantum mechanics, the concept of 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. h ∂ψ ( q , t ) i ¯ = H ¯ h ψ ( q , t ) ∂ t � is the Plank constant. H ¯ h is the quantized Hamiltonian: h ψ ( q , t ) := − � 2 H ¯ 2 m ∆ ψ ( q , t ) + V ( q ) ψ ( q , t ) where ∇ 2 is the Laplacian; h / ¯ h is the unit operator. ψ ( q , t ) = U ψ ( q , 0 ) , where U = e − iH ¯ 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 / ¯ h ψ ( q ) solves the Schrodinger equation, or equivalently, solve the Helmholtz equation − � 2 2 m ∇ 2 ψ ( q ) = E ψ ( 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 2 m ∇ 2 ψ ( q ) = E ψ ( q ) On the Hilbert space L 2 ( Q , m ) , Q compact and connected, the Laplacian is selfadjoint and has only discrete spectrum: − � 2 2 m ∇ 2 ψ n ( q ) = E n ψ n ( q ) The potential function imposes the Dirichlet condition: ψ n ( q ) = 0 , q ∈ ∂ Q Let λ n = 1 � 2 2 mE n . This induced to the Dirichlet-Laplacian problem: ∇ 2 ψ n ( q ) = − λ n ψ ( q ) , q ∈ Q , ψ n | ∂ Q = 0 √ λ n refers to the frequency of the wave function ψ n . Hong-Kun Zhang Quantum stadium and its spectral rigidity

Spectrum of quantized Hamiltonian Consider he Helmholtz equation − � 2 2 m ∇ 2 ψ ( q ) = E ψ ( q ) On the Hilbert space L 2 ( Q , m ) , Q compact and connected, the Laplacian is selfadjoint and has only discrete spectrum: − � 2 2 m ∇ 2 ψ n ( q ) = E n ψ n ( q ) The potential function imposes the Dirichlet condition: ψ n ( q ) = 0 , q ∈ ∂ Q Let λ n = 1 � 2 2 mE n . This induced to the Dirichlet-Laplacian problem: ∇ 2 ψ n ( q ) = − λ n ψ ( q ) , q ∈ Q , ψ n | ∂ Q = 0 √ λ n refers to the frequency of the wave function ψ n . Hong-Kun Zhang Quantum stadium and its spectral rigidity

Spectrum of quantized Hamiltonian Consider he Helmholtz equation − � 2 2 m ∇ 2 ψ ( q ) = E ψ ( q ) On the Hilbert space L 2 ( Q , m ) , Q compact and connected, the Laplacian is selfadjoint and has only discrete spectrum: − � 2 2 m ∇ 2 ψ n ( q ) = E n ψ n ( q ) The potential function imposes the Dirichlet condition: ψ n ( q ) = 0 , q ∈ ∂ Q Let λ n = 1 � 2 2 mE n . This induced to the Dirichlet-Laplacian problem: ∇ 2 ψ n ( q ) = − λ n ψ ( q ) , q ∈ Q , ψ n | ∂ Q = 0 √ λ n refers to the frequency of the wave function ψ n . 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: ∇ 2 ψ n ( q ) = − λ n ψ ( q ) , q ∈ Q , ψ n | ∂ Q = 0 The Dirichlet-Laplacian spectrum is denoted as ∆( Q ) , which contains all e-values 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ n → ∞ 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 ∆( Q 1 ) = ∆( Q 0 ) , then we say Q 1 and Q 0 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: ∇ 2 ψ n ( q ) = − λ n ψ ( q ) , q ∈ Q , ψ n | ∂ Q = 0 The Dirichlet-Laplacian spectrum is denoted as ∆( Q ) , which contains all e-values 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ n → ∞ 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 ∆( Q 1 ) = ∆( Q 0 ) , then we say Q 1 and Q 0 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: ∇ 2 ψ n ( q ) = − λ n ψ ( q ) , q ∈ Q , ψ n | ∂ Q = 0 The Dirichlet-Laplacian spectrum is denoted as ∆( Q ) , which contains all e-values 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ n → ∞ 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 ∆( Q 1 ) = ∆( Q 0 ) , then we say Q 1 and Q 0 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: ∇ 2 ψ n ( q ) = − λ n ψ ( q ) , q ∈ Q , ψ n | ∂ Q = 0 The Dirichlet-Laplacian spectrum is denoted as ∆( Q ) , which contains all e-values 0 < λ 1 ≤ λ 2 ≤ · · · ≤ λ n → ∞ 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 ∆( Q 1 ) = ∆( Q 0 ) , then we say Q 1 and Q 0 are isospectral. Hong-Kun Zhang Quantum stadium and its spectral rigidity

Can ∆( Q ) determines the shape of billiard table? Weyl’s Law (1911): 4 π λ n ∼ area ( Q ) · n M. Kac (1966) "Can you hear the shape of a drum?" n λ n t ∼ area(Q) − | ∂ Q | 8 √ π t + 1 − h � e 4 π t 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