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# Inverse Problems for Ultrahyperbolic Schr¨ odinger Equations 

Fikret G¨ olgeleyen and ¨ Ozlem Kaytmaz 

Department of Mathematics, Bulent Ecevit University, Zonguldak 67100 Turkey E-mail: f.golgeleyen@beun.edu.tr, ozlem.kaytmaz@beun.edu.tr 

## ABSTRACT 

Abstract. In this paper, we establish a global Carleman estimate for an Ultrahyper- bolic Schr¨ odinger equation. Moreover, we prove H¨ older stability for the inverse problem of determining a coefficient or a source term in the Ultrahyperbolic Schr¨ odinger equation by some lateral boundary data. 

Keywords. Ultrahyperbolic Schr¨ odinger equation, Inverse problem, Stability, Carle- man estimate 

Let n, m ∈ N , T > 0 and let D ⊂ R n be a bounded domain with smooth boundary ∂D and G = { y ∈ R $^{m}$; | y | < L } for L > 0 . 

We set Q = D × G × (0 , T ), Σ = ∂D × G × (0 , T ) and i = √ − 1 . 

We consider the ultrahyperbolic Schr¨ odinger equation 

## 1. INTRODUCTION 

i∂$_{t}$v ( x, y, t ) + Δ$_{y}$ v ( x, y, t ) − Δ$_{x}$ v ( x, y, t ) − p ( x, y ) v ( x, y, t ) = 0 , ( x, y, t ) ∈ Q, (1.1) 

with the following initial and Dirichlet boundary data 

<img src='content_image/319261.jpg'>

<img src='content_image/319262.jpg'>

