A spectral theorem for Sturm-Liouville operators with potentials that are finite sums of exponentials (Q1603134)

A finite sum of exponentials exp(igx) (over some \(K\) positive \(g\)'s) defines a potential \(V(x)\) and a Hamiltonian \(H\) which is not selfadjoint. The author proves that the set of the related (right) Hilbert-space eigenfunctions may still be understood as playing the same eigenfunction expansion role as their standard Sturm-Liouville analogs do in the selfadjoint cases. The precise meaning of this observation is formulated and proved as a theorem giving the expansion of a function \(f\) in terms of the eigenstates of \(H\) in the form of the Fourier-type pricipal-value integral over the spectrum. The author adds a remark which recommends an alternative arrangement of the eigenfunction expansion in the form that separates the contribution of the continuous spectrum from that of the isolated spectral singularities. Marginally, it is also worth noting that on the certain modified domains specified by the so-called \(PT\) symmetry requirement in quantum mechanics, the Hamiltonians \(H\) in question may still possess the pure point spectrum [cf. \textit{F. Cannata, G. Junker} and \textit{J. Trost}, Phys. Lett., A 246, No. 3-4, 219--226 (1998; Zbl 0941.81028) where \(K = 1\) and \textit{M. Znojil}, Phys. Lett., A 264, No. 2-3, 108--111 (1999; Zbl 0949.81020) where \(K=2\)] and could attract a further study motivated by the physics of bound states.