where A$_{1}$$_{,}$$_{2}$ and B$_{1}$$_{,}$$_{2}$ are functions of y$_{i}$ , defined via (55), and we have used (56) in (54) to obtain (57). The plan now is to perform a separation of variables in order to obtain a Sturm-Liouville eigenvalue problem, and then to analyye this eigenvalue problem to determine stability of the system. Because the 5D equations of motion of the scalar perturbations are coupled, the correct separation of variables ansatz is a coupled one where u ( n $^{)}$( x ) is the n th 4D Kaluza-Klein physical mode and Ψ ( n ) y ( y ) and ξ ( n ) y ( y ) are the wave functions. Plugging this ansatz into equations (52) and (53) leads to the following coupled equations The separation of variables thus yields a 4D wave equation for u ( x ) and a system of two coupled differential equations with boundary conditions for the profiles The system of equations (63) and (64) constitute an eigenvalue problem. The stability of the static background around which we have added scalar perturbations therefore depends on the existence, or absence, of a negative eigenvalue m 2 u associated with a solution to eqs. (63) and (64). This situation is somewhat unusual, since generally the Kaluza-Klein eigenvalue problem arising from dimensional reduction consists of a single second order differential equation, 23 Ψ ′ $_{y}$( y$_{i}$ ) = A$_{1}$ ( y$_{i}$ )Ψ$_{y}$( y$_{i}$ ) + A$_{2}$ ( y$_{i}$ ) ξ$_{y}$ ( y$_{i}$ ) (65) ξ ′ $_{y}$( y$_{i}$ ) = B$_{1}$ ( y$_{i}$ )Ψ$_{y}$( y$_{i}$ ) + B$_{2}$ ( y$_{i}$ ) ξ$_{y}$ ( y$_{i}$ ) . (66) D y $_{1}$Ψ$_{y}$ ( y ) − m 2 $_{u}$Ψ$_{y}$ ( y ) = D y $_{2}$ξ$_{y}$ ( y ) (63) D y $_{3}$ξ$_{y}$ ( y ) − m 2 $_{u}$ξ$_{y}$ ( y ) = D y $_{4}$Ψ$_{y}$ ( y ) (64) □ u ( x ) + m 2 $_{u}$u ( x ) = 0 (62) Ψ$_{y}$( y ) □ u ( x ) + u ( x ) D y $_{1}$Ψ$_{y}$ ( y ) = u ( x ) D y $_{2}$ξ$_{y}$ ( y ) (60) ξ$_{y}$ ( y ) □ u ( x ) + u ( x ) D y $_{3}$ξ$_{y}$ ( y ) = u ( x ) D y $_{4}$Ψ$_{y}$ ( y ) . (61) Ψ ( n $^{)}$( x, y ) = Ψ ( n ) y ( y ) u ( n $^{)}$( x ) (58) ξ ( n $^{)}$( x, y ) = ξ ( n ) y ( y ) u ( n $^{)}$( x ) , (59)