36 research outputs found

    An operator approach to indefinite Stieltjes moment problem

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    In the present paper we solve the indefinite Stieltjes moment problem MPkκ(s) within the M.G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A[0,N] generated by J[0,N]. The u-resolvent matrices of the operator A[0,N] are calculated in terms of generalized Stieltjes polynomials using the boundary triple’s technique. Criterions for the problem MPkκ(s) to be solvable and indeterminate are found. Explicit formulae for Pade approximants for generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented

    A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators

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    Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered.Comment: 38 pages, Proposition 2.2 and its proof corrected, Remarks 2.5, 3.4, and 3.12 extended, details added in subsections 2.3 and 4.2, section 6 rearranged, typos corrected, references adde

    Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

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    Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity

    Cantor and band spectra for periodic quantum graphs with magnetic fields

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    We provide an exhaustive spectral analysis of the two-dimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the Bethe-Sommerfeld conjecture fails in this case.Comment: Misprints correcte

    Application of Beta-Glucuronidase Transient Expression for Selection of Maize Genotypes Competent for Genetic Transformation

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    Genetic transformation of inbred maize lines and F1 hybrids registered in Ukraine has been carriedout. The study employed a biolistic method for genetic transformation of immature maize embryos thatformed callus tissue and the pAHC25 vector containing the genes of phosphinothricin-N-acetyltransferase(bar) and β-glucuronidase (uidA). As a result of the transformation of callus tissue of maize genotypes, linesresistant to phosphinothricin and regenerated plants were obtained. The activity of β-glucuronidase in herbicide-resistant calli was detected. The presence of the bar gene in callus DNA was demonstrated by the PCRmethod. The rate of stable transformation ranged from 2.2 to 30% depending on the genotype. The relationshipbetween the results of transient expression of the β-glucuronidase gene and stable genetic transformationwas observed. The proposed protocol for genetic transformation of maize using the study of transient expressionof the β-glucuronidase gene makes it possible to significantly simplify the process of selecting genotypescompetent for genetic transformation and create transgenic organisms with new traits

    A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

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    In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, SεIHS\geq \varepsilon I_{\mathcal{H}} for some ε>0\varepsilon >0 in a Hilbert space H\mathcal{H} to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for HK,ΩH_{K,\Omega}, the Krein--von Neumann extension of the perturbed Laplacian Δ+V-\Delta+V (in short, the perturbed Krein Laplacian) defined on C0(Ω)C^\infty_0(\Omega), where VV is measurable, bounded and nonnegative, in a bounded open set ΩRn\Omega\subset\mathbb{R}^n belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,rC^{1,r}, r>1/2r>1/2.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144

    A Factorization Model for the Generalized Friedrichs Extension in a Pontryagin Space

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    An operator model for the generalized Friedrichs extension in the Pontryagin space setting is presented. The model is based on a factorization of the associated Weyl function (or Q-function) and it carries the information on the asymptotic behavior of the Weyl function at z = ∞

    Asymptotic Expansions of Generalized Nevanlinna Functions and their Spectral Properties

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    Asymptotic expansions of generalized Nevanlinna functions Q are investigated by means of a factorization model involving a part of the generalized zeros and poles of nonpositive type of the function Q. The main results in this paper arise from the explicit construction of maximal Jordan chains in the root subspace R∞(SF) of the so-called generalized Friedrichs extension. A classification of maximal Jordan chains is introduced and studied in analytical terms by establishing the connections to the appropriate asymptotic expansions. This approach results in various new analytic characterizations of the spectral properties of selfadjoint relations in Pontryagin spaces and, conversely, translates analytic and asymptotic properties of generalized Nevanlinna functions into the spectral theoretical properties of self-adjoint relations in Pontryagin spaces
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