Librations of Hamiltonian systems and eigenvalue splitting inquantum double well

Anatoly AnikinBauman Moscow State Technical Unversity, anikin83@inbox.ru

International Conference on Applied Mathematics, Heraklion, September 16 –20, 2013

1

1 Energy splitting in the double well

Consider a Shrodinger operator in Rn

H = −h2

2∆ + V (x), h → 0.

We assume that(H1) V ≥ 0 , V ∈ C∞, V → ∞, x → ∞,(H2) V (x1, x2, . . . , xn) is symmetric with respect to the hyperplane

{x1 = 0} or to the origin x = 0,(H3) there are exactly two symmetric minima V (a±) = 0, and V ′′

xx(a±)has eigenvalues ω21, . . . ,ω

2n such that 2ω1 < min

j≥2ωj (ωj > 0).

(H4) Hamiltonian system H = K − V with standard kinetic energyK and the inverted potential −V satisfies some non-degeneracyconditions to be specified further.

V (x)

x1

x2, . . . , xn

V (x)

x

a+ a+a−a−

E±0

E±1

. . .

ψ(x)

x

ψ(x)

x

The spectrum of HE−0 , E

+0 , E

−1 , E+

1 , E−2 , E+

2 , . . .

is discrete and nearly twice degenerate. We are interested in expo-nentially small energy splitting E+

m − E−m (tunneling asymptotics).

V (x)

x

x

ψ+ν (x)

x

ψ−ν (x)

a− a+−c

Eν

−b cb

Actual eigenfunctions are odd and even.

The splitting problems:1. Splitting of the ground state. We calculate E+

0 − E−0 as h → 0.

2. Splitting of the low lying excited states. We calculate E+n − E−

nas h → 0, when n > 0 is fixed.

3. Splitting of the highly excited states. We calculate E+n − E−

n ash → 0, n → ∞, and E±

n → E0, where E0 > 0 is fixed.

2 1D case2.1 Highly excited states: Landau-Lifshitz formulaFix E > 0, in the O(h2)-vicinity of E as h → 0 we have two

eigenvalues, their difference is

∆E =ω0h

πe−

Sh(1 + o(1)), h → 0

S =1

2

∮

H=−E

p dx,

here ω0 is the frequency of motion H = E.Landau-Lifshitz, Fedoryuk ’65, Alenitsyn ’82.

p

V (x)

x

x

a− a+−c

E

−b cb

Λ+Λ−

p

a− a+−c

−E

−b cb−V (x)

x

xΓ

The classical Hamiltonian on the left. The tunnel Hamiltonianwith the inverted potential on the right.

2.2 The ground and low lying excited states: Harrell for-mulaFix m ∈ Z+, for the eigenstates

E±m =

ωh

2(2m + 1) + O(h2), E+

m − E−m = O(h∞)

we have

∆Em ≡ E+m − E−

m = bmωh

πe−

Sm,hh (1 + o(1)),

wherebm =

2−m√π

m!e12+m

, Sm,h =1

2

∮

H=−E±m

p dx.

Harrell ’78, Helffer-Sjostrand ’85.Note that bm → 1 as m → ∞.

2.3 The ground and low lying excited states: instanton formSplitting may be written down in a form:

∆Em = Am(h)e−S0h (1 + o(1)), h → 0

whereS0 =

∫

H=0

p dx.

p

V (x)

x

x

a− a+−c

Em → 0

−b cb

Λ+ → a+Λ− → a−

p

a− a+−c

−Em → 0

−b cb

−V (x)x

x

γ0 : H = 0

The instnton γ0 is a doubly asymptotic trajectory (the separatrix)of the tunnel Hamiltonian.

The amplitude has a cumbersome form. For the ground state, forinstance,

A0(h) = 2

√ωh

π|ϕ(0)| exp

∫ a2

0

(ω√2V

+ϕ′(z)ϕ(z)

)dz,

ϕ(x) = −

√

2

∫ a2

x

√2V (z) dz.

Slavyanov ’69, Pankratova ’84 Helffer-Sjostrand ’85.This asymptotics is easier to obtain than Harrell’s ones.

a+a−

x1

x2

3 Multidimensional case3.1 Additional hypothesis(H4) There is a unique doubly asymptotic trajectory γ0 connecting

unstable equilibria of the tunnel Hamiltonian a±, futhermore,1. γ0 approaches equilibria in a non-singular direction (i.e. associ-

ated with the lowest ω1) (Fig. 2).2. The asymptotic manifolds W±

± of the unstable equilibria a± in-tersect along γ0 transversally.

3.2 Instanton form of splittingFor the splitting of the ground state

E±0 =

h

2

n∑

j=1

ωj + O(h2), E+0 − E−

0 = O(h∞)

we haveE+0 − E−

0 = A0(h)e−S0

h (1 + o(1)). (1)Here S0 is action along the instanton γ0.1. S0 in the exponent: Maslov, Simon ’84.2. A0 ∼

√hB: Helffer-Sjostrand ’85.

3. Explicit formula for A0, where B in terms of variational equationsnear the instanton: Dobrokhotov, Kolokol’tsov, Maslov ’93.

3.3 Generalization for some excited statesThe lower part of the spectrum consists of

E±ν =

n∑

j=1

ωjh(2mj + 1)

2+O(h2),

where ν = (m1, . . . ,mn) is called quantum vector.We study the case ν = (m, 0, . . . , 0). Recall that ω1 is the lowest

frequency corresponding to the instanton direction.It is not hard to write out the correct formula for splitting. (Its

rigorous justification is much more difficult).

Main ideas:1. Use WKB-approximation for the quasimodes localized near a+

and a−:

ψ± = B±(x)e−S±(x)

h

2. S±(x) is action of asymptotic trajectory connecting x and a±.3. B±(x) is a solution of transport equation along the instanton.

Initial conditions are taken from harmonic oscillator approxima-tion.

4. Actual eigenfunctions are ψ1,2 = ψ+ ± ψ−.5. Integrate the identity

(E+m − E−

m)ψ1ψ2 = −h2

2(ψ1∆ψ2 − ψ2∆ψ1)

over the half-space x1 ≥ 0 and using Laplace method we expressE+m − E−

m via B± and derivatives of S± at the point, where γ0crosses x1 = 0. (Herring method)

4 Librations and splitting

a− a+

V = ε V = ε

In a neighborhood of γ0 there is a fam-ily of periodic trajectories, librations. SeeKozlov, Bolotin ’78.Normal form near the family:

H = H0(J)−n∑

j=2

βk(J)ujvj+R(J,ϕ, u, v

where J = Sπ is the "action variable" numbering librations, φ

mod 2π, and, u, v are transversal coordinates . Here H0(J) andβk(J) be respectively the energy and Floquet exponents of the as-sociated libration.

Theorem 1.

E+m − E−

m = bmω1h

πe−π

Jhh (1 + o(1)), h → 0,

where bm = 2−m√π

m!e12+m

and

H0(Jh)−h

2

n∑

k=2

βk(Jh) = −h

2

n∑

k=1

ωk.

This theorem is a mathematically rigorous version of an earlierresult:J. Bruning, S. Yu. Dobrokhotov, E. S. Semenov. Regul. Chaotic

Dyn. 2006. V. 11. No. 2. P. 167—180.See for the case of the ground state:A. Yu. Anikin. Rus. J. of Math. Phys. 2013. V. 20. No. 1. pp.

1–10.A. Yu. Anikin. Theoret. and Math. Phys. 2013. V. 175. No. 2.

pp. 609–619.

Comments

1. In the 1D case Theorem 1 becomes Harrell’s formula.2. In the case of the separating variables: V (x1, x2) = V1(x1) +V2(x2), where V1 is a double well, and V2 is a single well. Obvi-ously, β2 = ω2, and Jh is found from H0(Jh) = −ω1h

2 .3. In the proof we use only classical asymptotics. The key formula

is for the difference of actions along a libration and the instanton.We also express the sum of Floquet exponents via variationalequation.

4. It is interesting to interpret this theorem from the viewpoint ofthe dynamics in the complexified phase space.

5. It is unclear if this theorem may be generalized for quantum vec-tors different from (m, 0, . . . , 0)

Conclusion

1. Theorem 1 is the main result giving nice asymptotic formula forthe energy splitting of the ground state and some excited states(m, 0, . . . , 0).

2. In the case of the ground state Theorem 1 is proved rigorously.3. In the case of the excited states we derive Theorem from instanton

splitting formula. The latter requires proper justification.

Thanks for your attention!