# Brief Article

# Remark on the Kato smoothing effect for Schrödinger equation with superquadratic potentials

###### Résumé

The aim of this note is to extend recent results of Yajima-Zhang [Y-Z1, Y-Z2] on the - smoothing effect for Schrödinger equation with potential growing at infinity faster than quadratically.

## 1 Introduction

The aim of this note is to extend a recent result by Yajima-Zhang [Y-Z1, Y-Z2]. In this paper these authors considered the Hamiltonian where is a real and potential on satisfying for some and ,

(1.1) |

(1.2) |

and they proved the following. For any and one can find such that for all in ,

(1.3) |

where is the flat Laplacian.
In this note, using the ideas contained in Doï [D3] we shall show that one can handle variable coefficients Laplacian with time dependent potentials, one can remove the condition (1.2), one can replace the cut-off function in (1.3) by with any and finally that the weight is enough for the tangential derivatives.

When the estimate (1.3) goes back to Constantin-Saut [C-S], Sjölin [S], Vega [V], Yajima [Y] who extended to the Schrödinger equation a phenomenon discovered by T. Kato [K] on the KdV equation. Later on their results where extended to the variable coefficients operators by Doï in a series of papers [D1, D2, D3, D4] which contained the case of Theorem 1.1 below.

Let us describe more precisely our result. It will be convenient to introduce the Hörmander’s metric

(1.4) |

to which we associate the usual class of symbols if is a weight. Recall that iff and

If we shall set

(1.5) |

We shall consider here an operator of the form

(1.6) |

and we shall denote by the principal symbol of , namely

(1.7) |

We shall make the following structure and geometrical assumptions.

Structure assumptions. We shall assume the following,

(1.8) |

(1.9) |

(1.10) |

Geometrical assumptions. Let be the bicharacteristic flow of . It is easy to see that under the conditions (1.8), (1.9) it is defined for all . Let us set . Then we shall assume that,

(1.11) |

This is the so-called ”non trapping condition” which is equivalent to the fact that if then .

We shall consider and we set

(1.12) |

For let and be the Weyl quantized pseudo-differential operator with symbol .

Our first result is the following.

###### Theorem 1.1

Now even when is the flat Laplacian it is known that the estimate in the above Theorem does not hold with . However we have the following result. Let us set

(1.13) |

and let us denote by its Weyl quantization.

###### Theorem 1.2

Here are some remarks and examples.

###### Remark 1.3

1)We know that one can find and positive such that
, for all x in . Let . Then and since the operator is essentially self adjoint on . It follows that (1.9), (1.10), (1.11) and (1.14) are satisfied, therefore Theorem 1.1 and 1.2 apply. However the lower bound (1.2) assumed in [Y-Z2] is not satisfied.

2) Assume that with for some . Then if is small enough the non trapping condition (1.11) is satisfied.

## 2 Proofs of the results

Let us consider the symbol . A straightforward computation shows that under condition (1.8) one can find positive such that

(2.1) |

where denotes the Hamiltonian field of the symbol .

Then we have the following result due to Doï [D3].

###### Lemma 2.1

Assume moreover that (1.11) is satisfied then there exist and positive constants such that

(i) ,

(ii) , if is large enough.

The symbol is called a global escape function for . Here is the form of this symbol. Let be such that if , if and . With large enough and we have,

where

and if if . Details can be found in [D3].

Proof of Theorem 1.1

Let be such that in (where is a small constant chosen later on) and for . Following Doï [D3] we set,

(2.2) |

Then and we have

(2.3) |

Let be such that if if and . With given by Lemma 2.1 we set

(2.4) |

Finally we set

(2.5) |

where is an arbitrary small constant and a large constant to be chosen.

The main step of the proof is the following Lemma.

###### Lemma 2.2

(i) One can find such that for any there exist positive constants C, C’ such that

(2.6) |

(ii)

(iii)

Proof

First of all on the support of we have It follows that and Now

(2.7) |

where the ’s are defined below.

1) we have it is easy to see that
. Since on the support of

(2.8) |

2) we have By Lemma 2.1

(2.9) |

3) 2.3), (2.4) that . It follows from (

(2.10) |

4) . It follows from (2.2) that we have and since we have . It follows from Lemma 2.1 that . Now on the support of . Here we have

(2.11) |

5) 2.3) that . It follows from (

(2.12) |

We deduce from (2.10) and (2.12) that

Now . Since on the support of we deduce that . Taking and using the facts that , and on the support of we obtain

(2.13) |

6) we have ; this implies that . On the support of . We have

Therefore we obtain

(2.14) |

Gathering the estimates obtained in (2.8) to (2.14) we obtain

(2.15) |

Now on the support of we have so and using (2.15) we obtain (2.6). . Therefore writing

We use the symbolic calculus in the classes . We have , , so

By the symbolic calculus . Since we have on its support we will have

Finally .

End of the proof of Theorem 1.1.

Since we can set . Let us introduce . Then there exist absolute constants , such that . Now

Since and we obtain

By lemma 2.2 .

Now by Lemma 2.2 and the sharp Gårding inequality, we obtain

(2.16) |

On the other hand we have for any

(2.17) |

Using (2.16) and (2.17) with small enough, we obtain

Integrating this inequality between 0 and (in [0,T]) and using Gronwall’s inequality, we obtain the conclusion of Theorem 1.1.

Proof of theorem 1.2.

Let , if , if . Recall that according to (1.14) we have where and . Let us set

(2.18) |

Then we have the following result.

###### Lemma 2.3

Let be defined in Lemma 2.1. One can find positive constants , and such that if we set

(2.19) |

then

Proof

First of all we have

(2.20) |

Indeed we have and .

Let us set

(2.21) |

We claim that on the support of we have

(2.22) |

for some positive constants and .

Indeed a straightforward computation shows that

Since by Lemma 2.1 we have for and on the support of we deduce that when . When we have .