We further assume that there exists a constant r$_{0}$ > 0 such that and α > 0 is sufficiently small and 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 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. 5 3. KEY CARLEMAN ESTIMATE | R ( x, y, 0) | ≥ r$_{0}$, x ∈ D, | y | ≤ 2 L L > 1 √$_{α}$ max x ∈ D | x − x$_{0}$ | . ‖ f ‖ L 2 ( D ×{| y | <L − ϵ } ) ≤ C 2 ∑ k =1 ∥ ∂$_{ν}$∂ k t u ∥ θ L $^{2}$( ∂D + ×{| y | < 2 L }× ( − T,T ) . ϕ ( x, y, t ) = e γψ ( x,y,t $^{)}$, (3.1) ψ ( x, y, t ) = | x − x$_{0}$ | 2 − α | y − y$_{0}$ | 2 − β | t | 2 , (3.2) ∂D$_{+}$ = { x ∈ ∂D ; ( x − x$_{0}$ ) · ν ≥ 0 } (3.3)