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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 b 

Applied to (17), this implies 

the logarithm of which yields 

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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 

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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 

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