Tunnel effect, quantum resonances and microlocal analysis
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(2) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. Outline. Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits The geometrical setting Main result. ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(3) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. Breit-Wigner peaks In physics, the introduction of the notion of quantum resonance was motivated by the behavior of various quantities related to scattering experiments, as scattering cross sections, or time delays, … ▶. At certain energies, these quantities present peaks, which were modelized by a Lorentzian shaped function ▶. wa,b : λ 7→. 1 b/2 ( ) . π (λ − a)2 + b/2 2. b. ▶. Note that for ρ = a − ib/2 ∈ C, one has wa,b (λ) =. 1 Im ρ , π |λ − ρ|2. a. and the complex number ρ was called a resonance. ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(4) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. The Schrödinger equation A quantum particle of mass m, moving in a potential V ∈ C ∞ (Rn , R), is described by a solution of the time-dependent Schrödinger equation iℏ∂t ϕ(t, x) = P(x, ℏDx )ϕ(t, x), ▶. P(x, ℏD ) = − ℏ2 ∆ + V(x). x 2m which satisfies the normalization condition (∫ )1/2 ∥ϕ(t, .)∥L2 = |ϕ(t, x)|2 dx = 1. ϕ(t, x) is called the wave function of the particle, and |ϕ(t, x)|2 is its density of probability of presence at time t.. ▶. ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(5) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. ▶. A solution with separate variables t and x has to be of the form ϕ(t, x) = e−iEt/ℏ ϕ0 (x, ℏ),. where E ∈ R is an eigenvalue of the self-adjoint, unbounded operator P on L2 (Rn ), with eigenvector ϕ0 such that ∥ϕ0 ∥L2 = 1. b Now suppose Pψ0 = Eψ0 for some ψ0 (not in L2 ) and E = a − i ∈ C. 2 Its time evolution should be. ▶. 1. ψ(t, x) = e ℏ (−ita−tb/2) ψ0 (x), so that its density of probability of presence at time t is |ψ(t, x)|2 = e−bt/ℏ . |ψ0 (x)|2 ▶. The imaginary part b of E is the decay rate of that probability. ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(6) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. From eigenvalues to resonances: tunnel effect ▶ Suppose that a (1d) classical particle with energy E moves in one of the potential below. Its trajectory would be the same in both cases.. For a (1d) quantum particle, the situation is drastically different: there could be no L2 eigenfunction in the case at the left. ▶. E0. However the existence of a ”trapped set” will give rise to the existence of resonant states and corresponding resonant energies, or resonances. ▶. ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(7) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. Hamiltonian mechanics A classical particle with mass 1, when placed in the conservative force field F(x) = −∇V(x), follows the trajectory (x(t, x0 , ξ0 ), ξ(t, x0 , ξ0 )) given by { ẋ(t) = ξ(t), ˙ = −∇V(x(t)), ξ(t) ▶. where x0 , ξ0 are the position and the impulsion of the particle at t = 0. This is Newton’s Equation written in Hamiltonian form: In the phase space T∗ Rn , the trajectory is an integral curve of the vector field ▶. Hp = ∂ξ p∂x − ∂x p∂ξ , associated to the total energy of the particle p(x, ξ) = ξ 2 + V(x). ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(8) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. Quantum observables ▶. The Schrödinger operator on L2 (Rn ) is Pu(x) = −h2 ∆u(x) + V(x)u(x) ∫∫ (x + y ) 1 i(x−y)·η/h e p = Op(p)u(x) = , η u(y)dy dη. (2πh)n 2. ▶ To a real-valued classical observable a(x, ξ) (with suitable assumption), is associated a quantum observable Op(a), a well defined unbounded self-adjoint operator on L2 (Rn ). For example. ▶. qj (x, ξ) = xj. Op(qj ). = Mj : φ 7→ xj φ(x). pj (x, ξ) = ξj. Op(pj ). = hDj : φ 7→ hi ∂j φ. The famous uncertainty principle is nothing else but [Mj , hDj ] = ... Thierry Ramond. . ... . ... .. h · i. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(9) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. N. Bohr’s correspondence principle ”Quantum mechanics contains classical mechanics as a limit case. The point is to understand how this limiting phenomenon is achieved.” ▶. A nice example: Gutzwiller’s formula Let M be a compact manifold, and P a self-adjoint h-pseudodifferential operator on M, of principal type. Let γ ⊂ p−1 (0) a closed orbit of Hp , and Tγ its primitive period. For f such that supp f̂ ⊂ [−NTγ −C, NTγ +C]\{0}, we have ▶. tr f(P/h). =. N 1 ∑ eikSγ +iνγ,k π/2 Tγ f̂(−kTγ ) + O(h). 2π | det((dCγ )k − I)|1/2 k=−N. where Cγ is Poincaré first return map, Sγ the classical action along γ…. ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(10) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. Pseudo-differential operators Definition Let a ∈ S(T∗ Rn ). The Weyl quantization of a, is the operator on L2 (Rn ) given by ∫∫ 1 x+y Op(a)u(x, h) = ei(x−y)·η/h a( , η)u(y)dydη. (2πh)d 2 This definition extends to a’s in some symbol class, as for example Sδ (m), the set of smooth functions a such that. ▶. ∀α ∈ N2n , ∃Cα > 0, such that ∀h ∈]0, h0 ], |∂Xα ah (X)| ≤ Cα h−δ|α| ⟨X⟩m for some δ ∈ [0, 1/2] and some m ∈ R. ▶. If a ∈ S0 (0), then. ▶ Op(a). Op(a). is a bounded operator on L2 (Rn ).. h ◦ Op(b) = Op(c) where c = a#b = ab + {a, b} + O(h2 ). i ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(11) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. Microsupport. Definition Let (x, ξ) ∈ T∗ Rn . We say that u = 0 microlocally near (x, ξ) when Op(ψ)u. = O(h∞ ). ∗ n for some ψ ∈ C∞ 0 (T R ) with ψ = 1 near (x, ξ). The microsupport MS(u) ′ n of u ∈ S (R ) is the complement of the set where u vanishes microlocally. ▶. For example. MS(a(x)eiϕ(x)/h ). = {(x, ξ) ∈ T∗ Rn , a(x) ̸= 0, ξ = ∇ϕ(x)}.. ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(12) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. Microsupport of solutions of PDE’s If P = Op(p) is some pseudodifferential operator, and u ∈ S ′ (Rn ) is such that Pu = Eu, then ▶. MS(u) ⊂ p−1 (E). ▶. Much more precisely:. Theorem (Hörmander) Let u ∈ L2 (Rn ) a solution of Pu = 0, such that ∥u∥L2 ≤ 1. Let also (x0 , ξ0 ) ∈ T∗ Rn et T∗ < 0 < T∗ ∈ R such that t 7→ exp tHp0 (x0 , ξ0 ) is defined on ]T∗ , T∗ [. Then (x0 , ξ0 ) ∈ MS(u) ⇔ ∀t ∈]T∗ , T∗ [, exp tHp0 (x0 , ξ0 ) ∈ MS(u). Here P = Op(p) with p = p0 + o(1) as h → 0, with p0 real valued. ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(13) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. The geometrical setting Main result. Settings . ▶. We assume that V has a local non-degenerate maximum at 0 V(x) = E0 −. n ∑ λ2j j=1. with 0 < λ1 ≤ · · · ≤ λn .. 4. x2j + O(x3 ). Brief Article. Brief Article. The Author. The Author. November 6, 2012. November 6, 2012 {V (x) = E0 } V (x). 0. 2θ0. π(H). 0. ▶. We set. ( Fp = d(0,0) Hp =. 0 2Id 1 2 2 0 2 diag(λ1 , · · · , λn ). Notice that σ(Fp ) = {−λn , · · · , −λ1 , λ1 , · · · , λn }. ... Thierry Ramond. . ... . ... .. ). . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(14) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. The geometrical setting Main result. Stable/unstable manifolds ▶ We denote Λ− and Λ+ the stable incoming and outgoing manifold at (0, 0) respectively: { } Λ± = (x, ξ); exp(tHp )(x, ξ) → 0 as t → ∓∞. Λ− and Λ+ are smooth, Lagrangian manifolds, stable under the Hp flow, given near (0, 0) by. ▶. { } Λ± = (x, ξ); ξ = ∇φ± (x). with φ± (x) = ±. n ∑ λj j=1. ▶. 4. x2j + O(x3 ). For ρ± ∈ Λ± , one can see that ) ( Πx exp(tHp )(ρ± ) = g± (ρ± )e±λ1 t + O e±(λ1 +ε)t ... Thierry Ramond. . ... . ... as t → ∓∞. .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(15) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. The geometrical setting Main result. Assumptions (A1) The trapped set at energy E0 is K(E0 ) = {(0, 0)} ∪ H, with H = Λ− ∩ Λ+ \ {(0, 0)} ̸= ∅. H is the set of the homoclinic curves. (A2) ∀ρ− , ρ+ ∈ H, g− (ρ− ) · g+ (ρ+ ) ̸= 0. γ+ (ρ+ ). (0, 0) H γ− (ρ− ). ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(16) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. The geometrical setting Main result. Assumptions and notations We focus here on the particular case where ∪J (A3) H = j=1 γj , where γj are homoclinic trajectories ▶. (A4) The intersection Λ+ ∩ Λ− is transverse along each of the γj ’s. ▶. For j ∈ {1, . . . , J}, we denote: ▶ ▶. g± j the asymptotic directions of the trajectory γj , ∫ Aj = γj ξ · dx the classical action along γj ,. νj the Maslov index along γj . Morever, we set. ▶ ▶. 1∑ λj 2 j=2 n. E − E0 = hz,. L :=. and ζ = ζ(z) =. ... Thierry Ramond. . ... . ... .. 1 (L − iz). λ1. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(17) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. The geometrical setting Main result. A result . ▶. Consider the J × J matrix M = M(ζ, h) = (mjk )j,k where Γ(ζ + 21 ) mjk (ζ) = Kjk eiAj /h 1 , (iλ1 gj+ · gk− )ζ+ 2 √ √ λ1 Dj (t) 2π − π (νj + 1 )i k e( 2 −L)t 2 2 Kjk = e |g− | lim lim √ · λ1 t→−∞ ( 2 +L)t t→+∞ λ1 Dk (t) e. Theorem (Bony, Fujiie, R., Zerzeri) Let µ1 , . . . , µJ be the eigenvalues of M, and ΨRes (P) = {E ∈ C; ∃j ∈ {1, . . . , J} s.t. hζ µj (ζ, h) = 1}. For any small ϵ, we have, in B1 (ϵ) = Box (Ch, (Λ + λ1 − ϵ)h)\{Γ(h) + D(ϵh)}, ) ( h . dist(Res(P), ΨRes(P)) = o | log h| ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(18) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. The geometrical setting Main result. Theorem Let σ ∈ R. The pseudo-resonances E, such that are given by E = E0 + 2kπλ1. Re E. = E0 + σh + o(h),. ( h ) h h − iLh + iλ1 log µj +o , | log h| | log h| | log h|. for some k ∈ Z.. ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(19) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. The geometrical setting Main result. The. case of 1 homoclinic cure ▶ Let σ ∈ R. The pseudo-resonances E, such that Re E = E0 + σh + o(h), are given by ( ) ( ( ( h ) A) h σ) h E = E0 + λ1 2kπ − − i Lh − λ1 log m − i +o , h | log h| λ1 | log h| | log h|. for some k ∈ Z. E0 − Ch. E0. E0 + Ch. n 1∑ λj h 2 j=2. O. (. h | log h|. 2πλ1 | logh h|. ). ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
(20) Introduction: quantum resonances Classical and quantum mechanics Microlocal analysis Resonances associated with homoclinic orbits. The geometrical setting Main result. Strategy . Our results rely heavily on the study of what we call a microlocal Cauchy problem, of the form { (P − E)u = v microlocally in Ω, u = u− microlocally near S− , ▶. where Ω is some neighborhood of the trapped set, and S− is a suitable hypersurface in the incoming region of Ω. ▶. ▶. Our main resolvent estimate (and therefore the existence of a resonance free region) in some region of the complex plane of energies follows from a uniqueness result. The existence, and precise estimates on resonances follows from the existence of solution to the inhomogeneous microlocal Cauchy problem.. We obtain that way a Bohr-Sommerfeld like quantization condition for the resonances, which implies in particular that they accumulate on precisely defined curves as h → 0. ▶. ... Thierry Ramond. . ... . ... .. . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. ... Tunnel Effect, Quantum Resonances and Microlocal Analysis. . ... . ... . ... . ... ..
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