On the spectrum of the Schrödinger operators associated with two-particle system with point interaction

This dissertation investigates the spectral properties of one- and three-dimensional Schrödinger operators with singular perturbations modeled by delta and delta-prime potentials. In the one-dimensional case, we construct self-adjoint extensions $h_\lambda$ of the symmetric Laplace operator corresponding to interactions with two identical point potentials symmetrically located $at \pm x_0$. The essential spectra of the oprator $h_\lambda$ are determined, and conditions for the existence of negative eigenvalues are established. Furthermore, we derive the asymptotic behavior of the eigenvalues $z_u(\lambda)$ and $z_v(\lambda)$ as the potential parameter $\lambda$ approaches the critical values $\lambda_u$ and $\lambda_v$, respectively.
In the three-dimensional case, we investigate the point interaction problem for two arbitrary particles. Using the Krein–Visik–Birman extension theory in the momentum representation of the Hamiltonian, we construct the self-adjoint extension $H_\mu$ of the associated symmetric operator. It is shown that the essential spectrum of $H_\mu$ coincides with the non-negative real axis. We analyze the existence of negative eigenvalues of $H_\mu$, their dependence on the parameters $\mu$ and the interaction location $x_0 = (x_0,0,0) \in \mathbf{R}^3$, and derive the asymptotics of the eigenvalues $z_1(\mu)$ and $z_2(\mu)$ as they approach the threshold of the essential spectrum.