2019
Том 71
№ 11

Molyboga V. M.

Articles: 2
Article (Ukrainian)

Schrödinger Operators with Distributional Matrix Potentials

Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 657–671

We study $1D$ Schrödinger operators $L(q)$ with distributional matrix potentials from the negative space $H_{unif}^{− 1} (ℝ, ℂ^{m × m})$. In particular, the class $H_{unif}^{− 1} (ℝ, ℂ^{m × m})$ contains periodic and almost periodic generalized functions. We establish the equivalence of different definitions of the operators $L(q)$, investigate their approximation by operators with smooth potentials $q ∈ L_{unif}^{− 1} (ℝ, ℂ^{m × m})$, and also prove that the spectra of operators $L(q)$ belong to the interior of a certain parabola.

Article (English)

Singularly perturbed periodic and semiperiodic differential operators

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 785–797

Qualitative and spectral properties of the form sums $$S_{±}(V) := D^{2m}_{±} + V(x),\quad m ∈ N,$$ are studied in the Hilbert space $L_2(0, 1)$. Here, $(D_{+})$ is a periodic differential operator, $(D_{-})$ is a semiperiodic differential operator, $D_{±}: u ↦ −iu′$, and $V(x)$ is an arbitrary 1-periodic complex-valued distribution from the Sobolev spaces $H_{per}^{−mα},\; α ∈ [0, 1]$.