Seminars: Sh. Kh. Ergashova, Essential and discrete spectrum of the Schrödinger operator corresponding to systems of identical particles

SEMINARS


Seminar on Modern Problems of Complex Analysis (Sadullaev Seminar) January 15, 2026 12:00, Tashkent, National University of Uzbekistan, Room A304 (Department of Mathematics)

Essential and discrete spectrum of the Schrödinger operator corresponding to systems of identical particles Sh. Kh. Ergashova Samarkand State University
Abstract: We investigate the spectrum of the Schrödinger operator $h(k)$ describing a two-boson system on the three-dimensional lattice $\mathbb{Z}^3$. The operator features a potential $\hat v$ with support exactly given by the cylindrical set $D = \bigl\{ \mathbf{n} = (n_1, n_2, n_3) \in \mathbb{Z}^3 : \|n_1\| + \|n_2\| \le 1 \bigr\}$. We prove that for every $\lambda \in (-\pi, \pi]$, the operator $h(\lambda, \pi, \pi)$ possesses infinitely many eigenvalues below the essential spectrum. We also analyze the Schrödinger operator $H_\mu(K)$ for a three-fermion system on the one-dimensional lattice $\mathbb{Z}$, with $K \in \mathbb{T}$ denoting the total quasimomentum. In this model, particles interact only when occupying adjacent sites. For sufficiently large interaction strength $\mu$ and for total quasimomentum $K = 0$ and $K = \pi$, we establish that $H_\mu(K)$ has exactly one eigenvalue below the essential spectrum. Our results are obtained using methods from the spectral theory of self-adjoint operators in Hilbert spaces, elements of perturbation theory, and the technique of invariant subspaces. Website: https://us02web.zoom.us/j/8022228888?pwd=b3M4cFJxUHFnZnpuU3kyWW8vNzg0QT09