Small perturbations around the background ¯ φ ( y ) may be defined as in (9). The spectrum of these perturbations consists of solutions to the eigenvalue problem A useful form of this equation is obtained by performing a change of variables e σ ( y $^{)}$dy = dz and defining σ ( z ) = − 2 3 ln ( J ( z )) and W ( z ) = µ $^{2}$( z ) e − 2 σ ( y ( z )) to yield To proceed, we make use of the following integral inequality [45]. For any function f ( z ), such that f ( a ) = f ( b ) = 0, and with n nodes within the interval [ a, b ], there exists ρ ∈ R such that Applied to (17), this implies the logarithm of which yields which is a lower bound for the eigenvalues in terms of the background quantities σ ( z ) and µ ( z ) (which are contained in J and W ). In the case of the lowest eigenvalue we have and so a sufficient condition for perturbative stability ( m 2 0 ≥ 0) is We may formulate this explicitly in terms of the warp factor σ ( z ) so that finally, a static solution ¯ φ ( z ) of (14), obeying Dirichlet boundary conditions, is stable if 10 ∫ b a e − ρ [ − 3 $_{2}$σ $^{′′}$+ 9 $_{4}$σ ′ $^{2}$+ µ $^{2}$( z ) e − 2 $^{σ}$] dz ≤ e √ ρπ , (23) ∫ b a e − ρ [ J ′′ J + W ( z )] dz ≤ e √ ρπ . (22) m 2 0 ≥ 1 ρ ln( e √ ρπ ) − 1 ρ ln (∫ b a e − ρ [ J ′′ J + W ( z )] dz ) (21) m 2 n ≥ 1 ρ ln[( n + 1) e √ ρπ ] − 1 ρ ln (∫ b a e − ρ [ J ′′ J + W ( z )] dz ) (20) e ρm 2 n ∫ b a e − ρ [ J ′′ J + W ( z )] dz ≥ ( n + 1) e √ ρπ , (19) ∫ b a e − ρf $^{′′}$( z ) /f ( z ) dz ≥ ( n + 1) e √ ρπ . (18) ( Jϕ ) ′′ Jϕ − J ′′ J − ( W ( z ) − m 2 n ) = 0 . (17) ϕ$_{y}$ ( y$_{1}$ ) = ϕ$_{y}$ ( y$_{2}$ ) = 0 (15) ϕ ′′ y − 4 σ $^{′}$ϕ′ y − [ µ $^{2}$( y ) − m 2 $_{n}$e 2 σ ( y $^{)}$] ϕ$_{y}$ = 0 . (16)