On the Spectrum of the One-Particle Schrödinger Operator with Point Interaction

We consider a one-dimensional one-particle quantum system interacted by two identical point interactions situated symmetrically with respect to the origin at the points $$\pm x_{0}$$ . The corresponding Schrödinger operator (energy operator) is constructed as a self-adjoint extension of the symmetric Laplace operator. An essential spectrum is described and the condition for the existence of the eigenvalue of the Schrödinger operator is studied. The main results of the work are based on the study of the operator extension spectrum of the operator $$h$$ .