Spectral instability of some non-selfadjoint anharmonic oscillators

The purpose of this Note is to highlight the spectral instability of some non-selfadjoint differential operators, by studying the growth rate of the norms of the spectral projections Πn associated with their eigenvalues. More precisely, we are concerned with some anharmonic oscillators A(m,θ)=−d2dx2+eiθ|x|m with |θ|<min{(m+2)π4,(m+2)π2m}, defined on L2(R). We get asymptotic expansions for the norm of the spectral projections associated with the large eigenvalues of A(1,θ) and A(2k,θ), k⩾1, extending the results of Davies (2000) [4] and Davies and Kuijlaars (2004) [5].