Lower bounds on the eigenvalue gap for vibrating strings

We provide lower bounds on the eigenvalue gap for vibrating strings with fixed endpoints depending only on qualitative properties of the density function. For example, if the density ρ is symmetric on the interval [ 0 , a ] , and if λ 1 and λ 2 are the first two eigenvalues of u ″ ( x ) + λ ρ ( x ) u ( x ) = 0 in ( 0 , a ) with u ( 0 ) = u ( a ) = 0 boundary conditions, then λ 2 − λ 1 > max { 1 ∫ 0 a / 2 ( a 2 − x ) ρ ( x ) d x , π 2 ρ M a 2 } , where ρ M = max 0 ⩽ x ⩽ a ρ ( x ) . The ideas used also lead to applications in the case of monotone densities.