Browsing by Subject "Weyl function"

Now showing 1 - 3 of 3

Access status: Open Access ,
A general boundary value problem and its Weyl function
(2007) Ryzhov, Vladimir
We study the abstract boundary value problem defined in terms of the Green identity and introduce the concept of Weyl operator function $M(\cdot)$ that agrees with other definitions found in the current literature. In typical cases of problems arising from the multidimensional partial equations of mathematical physics the function $M(\cdot)$ takes values in the set of unbounded densely defined operators acting on the auxiliary boundary space. Exact formulae are obtained and essential properties of $M(\cdot)$ are studied. In particular, we consider boundary problems defined by various boundary conditions and justify the well known procedure that reduces such problems to the 'equation on the boundary' involving the Weyl function, prove an analogue of the Borg-Levinson theorem, and link our results to the classical theory of extensions of symmetric operators.
Access status: Open Access ,
Krein-von Neumann extension of an even order differential operator on a finite interval
(Wydawnictwa AGH, 2018) Granovs'kij, Âroslav Igorovič; Oridoroga, Leonid Leonidovič
We describe the Krein-von Neumann extension of minimal operator associated with the expression $\mathcal{A}:=(-1)^n\frac{d^{2n}}{dx^{2n}}$ on a finite interval $(a,b)$ in terms of boundary conditions. All non-negative extensions of the operator $A$ as well as extensions with a finite number of negative squares are described.
Access status: Open Access ,
On some extensions of the A-model
(Wydawnictwa AGH, 2020) Juršėnas, Rytis
The A-model for finite rank singular perturbations of class $\mathfrak{H}_{-m-2}\setminus\mathfrak{H}_{-m-1}$, $m \in \mathbb{N}$, is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces $(\mathfrak{H}_n)_{n\in\mathbb{Z}}$ admit an orthogonal decomposition $\mathfrak{H}^-_n \oplus \mathfrak{H}^+_n$, with the corresponding projections satisfying $P^{\pm}_{n+1}\subseteq P^{\pm}_n$, nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces.