Well-Posedness for Degenerate Schrödinger Equations

We consider the initial value problem for Schrodinger type equations 
$$\frac{1}{i}\partial_tu-a(t)\Delta_xu+\sum_{j=1}^nb_j(t,x)\partial_{x_j}u=0$$ 
with $a(t)$ vanishing of finite order at $t=0$ proving the well-posedness in Sobolev and Gevrey spaces according to the behavior of the real parts $\Re 
b_j(t,x)$ as $t\to0$ and $|x|\to\infty$. Moreover, we discuss the application of our approach to the case of a general degeneracy.


Decay conditions
When the coefficients b j (t, x) are pure imaginary we have well-posedness without loss of derivatives by the energy method and Gronwall inequality since H(t) is the sum of a self-adjoint operator and of a bounded operator in L 2 . We can not expect any kind of well-posedness for general real valued b j as one can check solving the equation

Decay conditions
When the coefficients b j (t, x) are pure imaginary we have well-posedness without loss of derivatives by the energy method and Gronwall inequality since H(t) is the sum of a self-adjoint operator and of a bounded operator in L 2 . We can not expect any kind of well-posedness for general real valued b j as one can check solving the equation by Fourier transform. Decay conditions as x → ∞ for the real parts b j (x) have been proved to be necessary in the case of a time-independent Hamiltonian H = ∆ x + n j=1 b j (x)∂ x j , Ichinose et al.

Sufficient conditions
Still in the time-independent case, the condition is sufficient for the well-posedness in

Sufficient conditions
Still in the time-independent case, the condition is sufficient for the well-posedness in 1−σ , σ < 1, Kajitani-Baba et al. These results are optimal. After minor changes, the same proof works for time-dependent H(t) provided that the coefficient a(t) of the Laplacian never vanishes so that | b j (t, x)| ≤ Ca(t) x −σ .

Sufficient conditions
Still in the time-independent case, the condition is sufficient for the well-posedness in 1−σ , σ < 1, Kajitani-Baba et al. These results are optimal. After minor changes, the same proof works for time-dependent H(t) provided that the coefficient a(t) of the Laplacian never vanishes so that | b j (t, x)| ≤ Ca(t) x −σ . As far as we know, there are no well-posedness results for time-dependent Hamiltonians with a(t) that may vanish.

Main Result Theorem
The Cauchy problem is well-posed in

Main Result Theorem
The Cauchy problem is well-posed in For any ≥ 0 and k = we have the same optimal spaces of well-posedness as in the time-independent case. Transforming iH(t) into a bounded from above operator We get the well-posedness of the Cauchy problem after performing a change of variable v (t, is an invertible pseudo-differential operator with symbol e Λ(t,x,ξ) , Λ(t, x, ξ) real-valued of order q, 0 ≤ q < 1. Transforming iH(t) into a bounded from above operator We get the well-posedness of the Cauchy problem after performing a change of variable v (t, is an invertible pseudo-differential operator with symbol e Λ(t,x,ξ) , Λ(t, x, ξ) real-valued of order q, 0 ≤ q < 1. We look for Λ(t, x, ξ) in order to establish the energy estimate v (t) L 2 ≤ C v (0) L 2 for any solution of the transformed equation Transforming iH(t) into a bounded from above operator We get the well-posedness of the Cauchy problem after performing a change of variable v (t, for any solution of the transformed equation The energy estimate (without any loss of regularity) follows by Gronwall's lemma if we find Λ such that

Introduction Strategy in the proof Change of variable Verification Concluding remarks and open problems
The crucial inequality for the symbol Λ We seek for a function Λ that solves for all |ξ| ≥ h, and such that ∂ t Λ(t, x, ξ) has the order 1 and a(t)∂ x j Λ has the order zero.

Introduction Strategy in the proof Change of variable Verification Concluding remarks and open problems
The crucial inequality for the symbol Λ We seek for a function Λ that solves for all |ξ| ≥ h, and such that ∂ t Λ(t, x, ξ) has the order 1 and a(t)∂ x j Λ has the order zero. This means that we make A(t) an operator of order 1 with negative principal symbol. In view of the sharp Gårding inequality, this leads to a bounded from above operator.

Introduction Strategy in the proof Change of variable Verification Concluding remarks and open problems
The crucial inequality for the symbol Λ We seek for a function Λ that solves for all |ξ| ≥ h, and such that ∂ t Λ(t, x, ξ) has the order 1 and a(t)∂ x j Λ has the order zero. This means that we make A(t) an operator of order 1 with negative principal symbol. In view of the sharp Gårding inequality, this leads to a bounded from above operator. As it is well-known, then the energy estimate gives the well-posedness in L 2 of the Cauchy problem for the operator S Λ .
are continuous, then we have (at least locally in time) a unique solution u ∈ C ([0, T ]; X ) of the original Cauchy problem for any given initial data u 0 ∈ X . The order of e Λ corresponds to the loss of derivatives and determines the space X .
are continuous, then we have (at least locally in time) a unique solution u ∈ C ([0, T ]; X ) of the original Cauchy problem for any given initial data u 0 ∈ X . The order of e Λ corresponds to the loss of derivatives and determines the space X . We obtain spaces of well-posedness from the following estimates:

Degeneracy leads to solvability in Gevrey spaces
In particular, when k = the operator e Λ is: • of a finite positive order δ for σ = 1, X is the Frechet space H ∞ (with a loss of δ derivatives);

Degeneracy leads to solvability in Gevrey spaces
In particular, when k = the operator e Λ is: • of order zero for σ > 1, X is the Banach space H m ; • of a finite positive order δ for σ = 1, X is the Frechet space H ∞ (with a loss of δ derivatives); • of infinite order described by the symbol e ξ 1−σ for σ < 1, X is the Frechet space H m,s , s = 1/(1 − σ).

Degeneracy leads to solvability in Gevrey spaces
In particular, when k = the operator e Λ is: • of order zero for σ > 1, X is the Banach space H m ; • of a finite positive order δ for σ = 1, X is the Frechet space H ∞ (with a loss of δ derivatives); • of infinite order described by the symbol e ξ 1−σ for σ < 1, X is the Frechet space H m,s , s = 1/(1 − σ). In this case, we have the same spaces of well-posedness as in in the time-independent case.

Degeneracy leads to solvability in Gevrey spaces
In particular, when k = the operator e Λ is: • of order zero for σ > 1, X is the Banach space H m ; • of a finite positive order δ for σ = 1, X is the Frechet space H ∞ (with a loss of δ derivatives); • of infinite order described by the symbol e ξ 1−σ for σ < 1, X is the Frechet space H m,s , s = 1/(1 − σ). In this case, we have the same spaces of well-posedness as in in the time-independent case. •• For k < the operator e Λ is of infinite order described by e ξ q , 0 < q < 1, q = q( , k, σ), even with a fast decay σ > 1. A strong degeneracy leads to well-posedness only in Gevrey classes of index s ≤ 1/q.

Solving modulo a prescribed order
Let us devote to the inequality which is to be satisfied by Λ. Let w (ξ) be a weight function corresponding to a possible order of solutions. It is sufficient to find λ(t, x, ξ) of the same order as that of w (ξ) such that

Introduction Strategy in the proof Change of variable Verification Concluding remarks and open problems
Solving modulo a prescribed order Let us devote to the inequality which is to be satisfied by Λ. Let w (ξ) be a weight function corresponding to a possible order of solutions. It is sufficient to find λ(t, x, ξ) of the same order as that of w (ξ) such that In fact, if we define Λ(t, x, ξ) by Λ(t, x, ξ) = (t)w (ξ) + λ(t, x, ξ), then we have a solution of still of the order of w (ξ) taking (t) such that (t) ≤ −K .

Absorbing lower order terms
It is natural to absorbe an error of the order of w (ξ) because terms of such an order appear under the principal part in the asymptotic expansion of the operator A(t) in any case. If w (ξ) is not of order zero, then we also need to control them in the application of the Gårding inequality.

Absorbing lower order terms
It is natural to absorbe an error of the order of w (ξ) because terms of such an order appear under the principal part in the asymptotic expansion of the operator A(t) in any case. If w (ξ) is not of order zero, then we also need to control them in the application of the Gårding inequality. The symbol of this part of the order of w (ξ) will be bounded by N| (t)| + N, N ≥ K , so we will choose (t) as a solution of (t) + N( (t) + 1) = 0, (t) > 0.

Introduction Strategy in the proof Change of variable Verification Concluding remarks and open problems
Splitting the phase-space The study of the inequality for λ(t, x, ξ) is crucial in the zone x,ξ since we have in the other part So, we can use here the above absorbing argument.

Introduction Strategy in the proof Change of variable Verification Concluding remarks and open problems
The solution for mild degeneracy The solution for mild degeneracy In this case we can take a time-independent solution λ 0 (x, ξ) where χ(y ) is a cut-off function.

Introduction Strategy in the proof Change of variable Verification Concluding remarks and open problems
The solution for mild degeneracy In this case we can take a time-independent solution λ 0 (x, ξ)

Introduction Strategy in the proof Change of variable Verification Concluding remarks and open problems
The order of the time-independent solution We have For = k the optimal choice of the order q, together with the related Gevrey index s < 1/q for q > 0, follows from The first line gives q = 1 − σ for σ < 1. Strong degeneracy -Splitting the extended phase-space For k < we split the extended phase-space (t, x, ξ) into two zones. Defining t ξ = ξ −(1−q)/(k+1) we introduce the • pseudo-differential zone: t ≤ t ξ ; evolution zone: t ≥ t ξ . We put in the construction of a solution λ(t, x, ξ) a first term which is localized to the pseudo-differential zone. The symbol λ ψ (t, ξ) is of order q by the definition of t ξ .

Introduction Strategy in the proof Change of variable Verification Concluding remarks and open problems
Strong degeneracy -Splitting the extended phase-space For k < we split the extended phase-space (t, x, ξ) into two zones. Defining t ξ = ξ −(1−q)/(k+1) we introduce the • pseudo-differential zone: t ≤ t ξ ; evolution zone: t ≥ t ξ . We put in the construction of a solution λ(t, x, ξ) a first term which is localized to the pseudo-differential zone. The symbol λ ψ (t, ξ) is of order q by the definition of t ξ . Taking a sufficiently large M it follows The solution in the evolution zone In the evolution zone we define λ e (t, x, ξ) = λ e,0 (t, x, ξ) + λ e,1 (t, ξ) where λ 0 (x, ξ) is the time independent solution for k = and the weight function

Fixing the order
We have a solution in the evolution zone as soon as ∂ t λ e (t, x, ξ) ≤ 0. We have this fixing a large constant C 1 in the correction term λ e,1 (t, ξ).

Fixing the order
We have a solution in the evolution zone as soon as ∂ t λ e (t, x, ξ) ≤ 0. We have this fixing a large constant C 1 in the correction term λ e,1 (t, ξ). Using the definitions of t ξ and the order of the time-independent term λ 0 (x, ξ), the symbol λ e (t, x, ξ) can be estimated by

Fixing the order
We have a solution in the evolution zone as soon as ∂ t λ e (t, x, ξ) ≤ 0. We have this fixing a large constant C 1 in the correction term λ e,1 (t, ξ). Using the definitions of t ξ and the order of the time-independent term λ 0 (x, ξ), the symbol λ e (t, x, ξ) can be estimated by In order to have also λ e of order q (or q log for σ = 1) we choose q = Localizing the support of λ(t, x, ξ) for |ξ| ≥ h with a sufficiently large h, we can make the change of variable v = e Λ u invertible with (e Λ ) −1 given by a Neumann series of operators. Then, for the transformed operator S Λ we have This gives the energy estimate without loss of derivatives hence the well-posdness in L 2 of the Cauchy problem for S Λ . Localizing the support of λ(t, x, ξ) for |ξ| ≥ h with a sufficiently large h, we can make the change of variable v = e Λ u invertible with (e Λ ) −1 given by a Neumann series of operators. Then, for the transformed operator S Λ we have This gives the energy estimate without loss of derivatives hence the well-posdness in L 2 of the Cauchy problem for S Λ . Taking the order of e Λ into account (the transformation carries the loss) we have the results of well-posedness for the operator S.

General degeneracy
Let us consider a general coefficient a(t) increasing and such that a(t) = 0 (also of infinite order) and let us assume with µ(t) decreasing and such that lim t→+0 µ(t) = ∞. The separating line t = t ξ in the extended phase-space is now defined by In the definition of λ e (t, x, ξ) the factor t k− is replaced by µ(t).
The operator P can be formally factorized in the product of two (pseudo-differential) Schrödinger operators Pu := u tt + a 2 (t)∆ 2 with a(t) ≥ 0 vanishing at t = 0 of finite order and with real-valued b α (t, x) with |α| = 3 satisfying |b α (t, x)| ≤ Ct j x −σ with ≤ j < 2 .
The operator P can be formally factorized in the product of two (pseudo-differential) Schrödinger operators modulo terms of order 2. Performing a complete diagonalization in the evolution zone one should obtain for P the same results as for S with k = j − .

Necessity
An interesting problem is the optimality of the results, a subject widely studied for non-degenerate models. One can not find "better" spaces of well-posedness in the case = k in view of the necessary decay conditions as x → ∞ obtained for = k = 0. We conjecture to have sharp spaces of well-posedness also for k < . In particular, even assuming the strongest decay rate σ > 1 we conjecture that the Cauchy problem is not well-posed neither in H ∞ nor in H ∞,s for s > ( + 1)/( − k) in this case.