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We further assume that there exists a constant r$_{0}$ > 0 such that 

and α > 0 is sufficiently small and 

<img src='content_image/316638.jpg'>

<img src='content_image/316639.jpg'>

Then for any small ϵ > 0, for all real valued f ∈ L $^{2}$( D × {| y | < L } ) there exist constants C > 0 and θ ∈ (0 , 1) depending on ϵ, M, x$_{0}$, such that 

<img src='content_image/316640.jpg'>

## 3. KEY CARLEMAN ESTIMATE 

In this section, we show a Carleman estimate for the ultrahyperbolic Schr¨ odinger equation which will be used in the proof of our main result. 

Let Ω = D × G × ( − T, T ) , Γ$_{x}$ = ∂D × G × ( − T, T ), Γ$_{y}$ = D × ∂G × ( − T, T ) and let ∂ Ω = Γ$_{x}$ ∪ Γ$_{y}$ . 

Let x$_{0}$ / ∈ D, y$_{0}$ ∈ R m and α, β ∈ (0 , 1) , we set the weight function 

where 

γ > 0 is a parameter. Moreover, we set the geometric condition 

for x$_{0}$ / ∈ D. 

<img src='content_image/316642.jpg'>

<img src='content_image/316643.jpg'>

<img src='content_image/316641.jpg'>

