Spectral singularities of non-Hermitian Hamiltonians and SUSY transformations

Simple examples of non-Hermitian Hamiltonians with purely real spectra defined in L(R) having spectral singularities inside the continuous spectrum are given. It is shown that such Hamiltonians may appear by shifting the independent variable of a real potential into the complex plane. Also they may be created as SUSY partners of Hermitian Hamiltonians. In the latter case spectral singularities of a non-Hermitian Hamiltonian are ordinary points of the continuous spectrum for its Hermitian SUSY partner. Conditions for transformation functions are formulated when a complex potential with complex eigenenergies and spectral singularities has a SUSY partner with a real spectrum without spectral singularities. Finally we shortly discuss why Hamiltonians with spectral singularities are ‘bad’. Submitted to: J. Phys. A: Math. Gen. There are two essential differences between non-Hermitian Hamiltonians having a purely real spectrum and Hermitian Hamiltonians. The first difference is related to the discrete spectrum and consists in the possibility of appearance of associated functions [1, 2] (also called in [3] ‘background eigenfunctions’), which are not eigenfunctions of the Hamiltonian but they should be added to the set of discrete spectrum eigenfunctions to complete a basis in corresponding Hilbert space. Hamiltonians of this kind are known as non-diagonalizable. In my previous Letter [3] it was shown that they can be transformed into diagonalizable forms by appropriate SUSY transformations. In this note I show that SUSY transformations may be useful to ‘cure’ another ‘disease’ of non-Hermitian Hamiltonians which is related just to the second difference between Hermitian and non-Hermitian Hamiltonians consisting in appearance of spectral singularities inside a continuous spectrum. The paper by Naimark [4] was one of the first most essential contribution to the spectral theory of a non-selfadjoint operator of Schrödinger type. In particular, he was the first who noticed the possibility of appearance of spectral singularities inside a continuous spectrum. Later Lyantse [5] (see also [2]) studied carefully some properties of Hamiltonians with spectral singularities. In this note using the simplest real nontrivial potential without a discrete spectrum we demonstrate that the usual practice of getting exactly solvable complex potentials with a purely real spectrum consisting in simple shifting of the dependent variable of a real potential to the complex plane may lead to a potential with spectral singularities. Then we show how a spectral singularity may appear after a SUSY transformation over a real potential and how it can be ‘removed’ (more precisely it can be transformed into an ordinary point of Spectral singularities of non-Hermitian Hamiltonians and SUSY transformations 2 the continuous spectrum of a SUSY partner Hamiltonian). Finally we shortly discuss why Hamiltonians possessing spectral singularities are ‘bad’. For simplicity we will consider Sturm-Liouville problems on the positive semiaxis since in this case the continuous spectrum is non-degenerate and spectral singularities are easier ‘to handle’. Nevertheless, we would like to point out that interested reader can find a deep study of nonHermitian Hamiltonians defined on the whole real line, which includes the strict formulation and proof of the inverse scattering theorem, in [6]. (We notice that spectral singularities constitutes a part of spectral data.) We start with the definition of spectral singularities of a non-Hermitian one-dimensional Hamiltonian mainly following the paper by Lyantse [5]. Let us have a complex-valued function V (x) such that ∫ ∞