On the spectrum of the two-particle Schrödinger operator with point potential: one dimensional case

In the paper, a one-dimensional two-particle quantum system interacted by two identical point interactions is considered. The corresponding Schrödinger operator (energy operator) $h_\varepsilon$ depending on $\varepsilon$ is constructed as a self-adjoint extension of the symmetric Laplace operator. The main results of the work are based on the study of the operator $h_\varepsilon$. First, the essential spectrum is described. The existence of unique negative eigenvalue of the Schrödinger operator is proved. Further, this eigenvalue and the corresponding eigenfunction are found.