Tunneling for a semi-classical magnetic Schrödinger operator with symmetries

We are interested in decay estimates of the ground state (or the low energy eigenstates), outside the potential wells, for a semi-classical Magnetic Schrödinger operator with smooth coefficients PA (x, hD x ) = (hD x − µA(x))2 + V (x) on L 2(R d ). We shall essentially consider the case where µ is large. This kind of estimates, in case of Schrödinger operator without a magnetic field, have been studied by Agmon [1], also in the case of a Riemannian manifold M. Agmon estimates hold true for any h, but are particularly useful in the limit h → 0 when studying tunneling.

1. Agmon estimates and tunneling parameters for the ground states Given a semi-classical Schrödinger operator P 0 (x, hD x ) = −h 2 ∆ + V (x) on L 2 (M ), M a smooth manifold, we consider spectral effects induced by adding a magnetic potential (minimal coupling).For simplicity we assume here that P 0 and P A have compact resolvent.
A lot is known on the mathematical theory of magnetic classical and quantum Hamiltonians : see e.g.[15], [16] and [8], [9] with references therein.The link between the classical flow and the spectral asymptotics, e.g.trace formulas, is examined in [15], [16].This does not require precise information on eigenfunctions.Estimates on eigenfunctions and WKB solutions considered in [8], [9] are in turn of special importance for investigating tunneling between "potential wells", related to the decay of eigenfunctions.The fact that eigenfunctions are real or complex also plays a role.Within our framework (an application is given in [14]), we can summarize some of the main problems related to Magnetic Schrödinger operators P A (x, hD x ) = (hD x − µA(x)) 2 + V (x) as follows.
• Problem No.1: Compare the first (non degenerate) eigenvalue λ 0 (h) of P 0 (x, hD x ) with the first eigenvalue λ A (h) (possibly degenerate) of P A (x, hD x ).
Kato inequality shows that λ 0 (h) ≤ λ A (h).More refined properties rely on the topology of M : If M is an open bounded subset of R d with smooth boundary (non necessarily simply connected), then λ 0 (h) = λ A (h) iff dA(x) = 0 in M and the cohomology class of A verifies [A] ∈ H 1 (M, 2πhZ), see [17], [7].This reminds of Aharonov-Bohm effect when ), in which case P A has continuous spectrum.Another remarkable result (which is not to be used here) gives an upper bound on λ A (h). Namely, if M is a compact manifold without boundary, then λ A (h) is bounded by Mañé constant for the Lagrangian L A associated with The minimizer φ is of Lipschitz class.This is investigated within "weak KAM" or "weak WKB" theory.We have also the bound ), expressing in particular the gauge invariance of the spectrum, see [21].
• Problem No.2: Compare the decay of u A (h) with this of u 0 (h) > 0. Generally, λ A (h) is simple, but u A (h) is complex and may have zeroes (vortices in 2-D, threads in 3-D, . . .).When λ 0 (h) = λ A (h) as above, then u A (x; h) = u 0 (x; h)e iµ( x A(y) dy)/h , so the magnetic potential with dA(x) = 0 induces no additional decay.
Let M be an open subset of R d with smooth boundary, and ≤ E} be the potential well in M at energy E. First we assume that U E is connected.
We focus to the case where V (x) ≥ E = 0 and U E = {x ∈ M : V (x) = E} = {x 0 } is a non degenerate minimum (simplest case).Actually energy level E = 0 will be lifted by λ 0 (h) = O(h).
Decay estimates in some global L 2 norm with exponential weight e Φ 0 (x)/h for an eigenstate u 0 (h) of P 0 outside U E are known as Agmon estimates.They hold in any Sobolev norm and hence also pointwise.
Let us add the vector potential A, with A(x 0 ) = 0, and assume ρ 0 = (x 0 , 0) ∈ T * M is a non degenerate elliptic point for P A (x, ξ), that we call the magnetic well.
In case of P A we know already that Agmon type estimates with the same weight e Φ 0 (x)/h hold for u A (h) [13], [7].Thus we call them "basic Agmon estimates".But generally speaking, magnetic fields are confining, so we can expect that the eigenfunction u A (h) will decay faster indeed than u 0 (h) outside x 0 .
Local WKB constructions of u A (h) near the magnetic well, using quadratic approximation near the elliptic point, are computed in [20].When V, A are analytic, they are of the form and u A (h) decays generally faster than u 0 (h) in B A (x 0 , r 0 ).Using almost analytic extensions, we can show that (1) holds true when A and V are merely C ∞ , provided we replace O(e −ε A /h ) by O(h ∞ ).However, it is difficult to extend these expansions in the larger domain M .To this end, we introduce here some relative Agmon estimates: namely, at least for sufficiently large coupling constant µ, u A (h) decays in L 2 (M ) norm as e −Φ 1 (x)/h with Φ 1 > Φ 0 , and Φ 1 ≈ ReΨ A in B A .
• Problem No.3 (tunneling for the double well {x < , x > }).When M = R d instead, we assume that P 0 and P A commute with some hyperplane symmetry σ( In the presence of magnetic wells, we are interested in an estimate (from above) of the splitting between the two first eigenvalues λ ± A (h), which, incidentally, may coincide.Much is known for the splitting (from above and from below) between λ + 0 (h) > λ − 0 (h), see [12], [18].
In case V has non-degenerate minima at x >,< ∈ {±x d > 0}, V (x >,< ) = E = 0 we can construct (quite accurately from the point of vue of tunneling) approximate eigenfunctions u ± 0 (h) for P 0 in L 2 (R d ) from the ground state u 0 (h) of the one-well problem, i.e. localized at x 0 = 0.Here u − 0 (h) is even in x and given roughly by Here we have assumed for simplicity that u 0 is even in x, see [6] for the correct formula.[3], [2].An instanton γ is a limiting curve of the family of librations, as h → 0, i.e. as U >,< (h) shrinks to {x >,< }, the well at energy 0. This is a minimal geodesic for Agmon distance between x < and x > at energy 0. Instantons form generically a discrete set.

Define as a libration γ(h) a minimal geodesic for Agmon distance at energy λ
Localizing Agmon estimates near γ gives [12], [18]: for all δ > 0 (here we use the stable/unstable manifold theorem).We shall content ourselves with leading exponential estimates, allowing for arbitrary small loss δ in the exponent.Introduce now the magnetic potential.Again the localization procedure works, and it suffices to know the ground state u A (h) of the one-well problem.
For very small (h dependent) coupling constant µ, WKB expansions of u A (h) yield an estimate similar to (2) on the splitting ∆λ A (h) [13].For small coupling constant µ (but independent of h) we need analyticity of V, A to ensure exponential accuracy on WKB solutions.
Under some suitable hypotheses, we expect that our relative Agmon estimates approximate ∆λ A (h) in a similar way to (2), and moreover This holds actually in 2 regimes: (1) the coupling constant µ is sufficiently large, so that the decay of u A (h) is little sensitive to V ; (2) µ 2 |A(x)| 2 − V (x) is positive and small enough.
Recent results in the 2-D case of compactly supported and radially symmetric potential wells, and constant B ( [4], [10]), express the rate of exponential decay in term of the hopping formula.See also [11], and [5] when V = 0.

Precise statements and outline of proofs 1) General Agmon identities in the case of a magnetic potential
The scalar and magnetic potentials V, A will be assumed to be C ∞ .Let Ω ⊂ R d be a bounded open set with C 2 boundary.
We will consider weighted integrals, related with Dirichlet forms, like Ω |(hD x − µA(x))e Φ(x)/h u| 2 dx or Ω |(hD x − µA(x) + ihf (x))e Φ(x)/h u| 2 dx for a suitable real vector field f .When u is a complex C 2 function on Ω, we define the real vector field (quantum current with U (1) symmetry) It is not a priori integrable on Ω when Ω is not simply connected but if u is not vanishing, then Using Green formula one can show: where ∇Φ ∈ L ∞ (Ω) is understood in the usual sense, and dσ is the surface measure on ∂Ω.
As a particular case Φ I = 0, we retrieve the classical formula [7], p.95 which will give the "basic" Agmon estimate :
For the symmetric double well problem in a "large" bounded domain M ⊂ R d , with a single well U >,< (h) on each side of the hyperplane x d = 0. consider instead Dirichlet realization of P 0 (x, hD x ) = −h 2 ∆ + V (x) , containing sub-domains M >,< ⊂ M as above symmetric of each other with respect to We set Φ 0 (x) = min d 0 (x, U > (h)), d 0 (x, U < (h)) .Then (7) holds true for Φ 0 (x) instead of Φ 0 (x), see also [12], [7] when M = R d .

3) Standard Agmon estimates with a magnetic potential
Consider next the Dirichlet realization of P A (x, hD x ) = (hD x −µA(x)) 2 +V (x) in the domain M ⊂ R d , λ A (h) its ground state energy (possibly degenerate), and u A (h) an eigenfunction associated with λ A (h).As before we start with the "magnetic one-well problem".Let Ω be the complement of a (euclidian) ball B A ⊂ M (the projection of the "magnetic well").
We have the "basic" Agmon estimate for u A (x; h), which we express as in (7).Namely for all δ > 0, there is C δ > 0 such that, uniformly for 0 At this point we use the precise local decay estimates (1) of u A (h) to control the RHS of (8).Recall Ψ A is a (possibly complex) phase with real part that we shall assume strictly larger than Φ 0 on ∂B A , at least for large µ.This assumption is supported by the following example on L 2 (R 2 ) with ground state of the form u A (x; h) = Const.exp[−c(µ)x 2 /2h], and ground state energy λ A (h) = c(µ)h = (1 + µ 2 /4) 1/2 h.Actually this extends easily to higher dimensions, see [22].After some manipulation, (8) reduces to an estimate like (7) where the RHS can be replaced by O(e − ε 1 h ), for some ε 1 > 0, provided u A (x; h) decreases indeed faster (as in the latter example) than e −Φ 0 (x)/h on ∂B A .

4) R.Lavine and M.O.Carroll's formula and relative Agmon estimate
Contrary to (5) that suggests to introduce Agmon distance d 0 as a natural geometric guideline, (4) encodes no direct geometric information for complex Φ.This reflects the intrication between the cyclotronic motion and the magnetic drift at the classical level.We only know [16] that these motions are relatively well decoupled for large µ.Thus we can expect that in this regime to factor out the wave function as u A (h) = u 0 (h)v(h) and find Agmon type estimates on v(h).Recall the following formula [17]: Let A, f be real vector fields locally square integrable on R d , then for all u ∈ C ∞ 0 (R d ; C) we have Let (λ 0 (h), u 0 (h)) be the ground state of the Dirichlet realization of P 0 in M , then f = ∇u 0 /u 0 is a natural choice, for Let as above λ A (h) (we assume to be non degenerate) be the ground state energy of of P A (x, hD x ) on L 2 (M ) with Dirichlet boundary condition, and u A (h) the corresponding normalized eigenfunction, which we write as u A (h) = u 0 (h)v(h).Suitably modifying (10) with u = e Φ/h u A (h) to account for boundary terms on Ω ⊂ M , we can express dx and get : To get rid of the extra term on the RHS we try to solve This can be done if , and arg u 2 A (x; h) is a smooth function.Vanishing of u A maybe an obstruction for solving (12), in particular when the zeroes x j of J v (x) are vortices.We can also try to solve (12) approximately along a magnetic line γ, namely look for Φ I such that the 1-form 2dΦ I (x) − µA(x) vanishes on γ.
Once the contribution of J v (x) in (12) has been removed, we are left with an expression similar to (5), with |A(x)| 2 instead of V (x).Provided A = 0 outside the well, we solve an eikonal inequality and put Φ R = (1 − δ)Φ 1 .This yields again an Agmon estimate of the form (8), with Φ 1 instead of Φ 0 .On the other hand, we know that u 0 (x; h) is everywhere of the same order of magnitude as e −Φ 0 (x)/h .More precisely, if 0 ≤ V ∈ C ∞ (M ) and u 0 be the normalized ground state of P 0 = −h 2 ∆ + V (x) on L 2 (M ), then for all δ > 0, u 0 (x; h) ≥ C δ e −(d 0 (x,U h )+δ)/h locally uniformly on any compact set K ⊂ M \ U 0 (h), see e.g.[19]; moreover in a neighborhood of the quadratic well x 0 , we have |u 0 (x; h)| ≤ Ch −N e −Φ 0 (x)/h (as well as all derivatives), see [7], Proposition 3.3.5.In turn, this gives again an estimate like (7), so there is a set of two (independent) Agmon estimates for u A (h).Let Φ A (x) = max(Φ 0 (x), Φ 1 (x)), which amounts to consider the (Lipschitz)

Thus we proved:
Theorem 4: Let u A (h) be an eigenfunction of P A on L 2 (M ) with eigenvalue λ A (h). Assume we have solved (12).Then with a(δ) → 0 as δ → 0.Moreover, if the real part of Ψ A (x) = Ψ A (x, µ) is increasing with µ on ∂B A (x 0 , r 0 ) as in Example (9), then the remainder term O(e 2a(δ)/h ) can be improved to O(e − ε 2 /h ) for µ large enough, ε 2 > 0. This is consistent with (9) where But it is clear that our method cannot capture the precise decay of (9).
The point is that in general d A is no longer a (degenerate) Riemannian metric.This situation is met however in the following cases, discarding the energy shifts λ 0 (h) or λ A (h) − λ 0 (h) in the eikonal inequalities ( 6) and ( 13) : (1) Φ A = Φ 1 almost everywhere (i.e. the magnetic potential is more confining than the scalar potential); (2) the potentials verify µ 2 |A(x)| 2 = V (x), or µ 2 |A(x)| 2 − V (x) is sufficiently small.The latter case can be treated by perturbation theory, since the trajectories for ξ 2 − V (x) at energy λ 0 (h) and ξ 2 − |A(x)| 2 at energy λ A (h) − λ 0 (h) will be comparable, in particular near the set of minimal geodesics of either system.

Application to tunneling
We present here the main ideas towards (3).As usual [13] we need first an assumption on the spectrum (spectral gap).Let λ ± A (h) be the two first eigenvalues of on the localized domains M = M >,< as above.Assume µ A (h) is simple and asymptotically simple, see e.g.[19].As a rule, we expect the splitting ∆λ A (h) = λ + A (h) − λ − A (h) to be O(e −S A /h ) for some "Agmon distance" S A > 0 between the magnetic wells, to be computed in term of the ground state u >,< of P M >,< A , see [13], Theorem 3.1 and Remark 3.7.The "gap formula" is thus of the form ∆λ A (h) = h 2 Γ u > (y)∂ n u < (y) − u < (y)∂ n u > (y) dS(y)+ h Γ u > (y) A(y), n(y) u < (y) − u < (y) A(y), n(y) u > (y) dS(y) (15) where Γ ⊂ {x d = 0} is a "geodesic bisector" and n a normal to Γ.The eigenvalues λ ± A (h) are exponentially close to the corresponding µ A (h), and the functions u > , u < are related by the hyperplane symmetry.
In the case Φ A = Φ 1 , we apply Theorem 4 to the functions u > , u < , which allows to bound from above the first two traces of u > , u < in L 2 (Γ) with the exponential weight e −Φ A /h .This is essentially done as in the situation without the magnetic potential.This gives ∆λ(h) = O(e −(S A −c)/h ), c arbitrary small, where S A is (ordinary) Agmon distance between the wells x > and x < for the potential |A(x)| 2 .
The case ε W (x) = µ 2 |A(x)| 2 − V (x) small (where we can set µ = 1) is more difficult since we must work directly from (11).We can replace Ω by a neighborhood Ω 1 of a minimal geodesic between x > and x < common to |A(x)| 2 and V (x) when they are equal, and use perturbation theory in ε.We expect again ∆λ A (h) = O(e −(S A −c)/h ), c arbitrary small, where S A is, at 0-order in ε, Agmon distance between the wells x > and x < for the potential |A(x)| 2 = V (x).Moreover when W (x) > 0, Φ R should be strictly larger than Φ 0 in Ω 1 , which implies (3).