Теория Титчмарша - Вейля сингулярного уравнения Хана - Штурма - Лиувилля

In this work, we will consider the singular Hahn--Sturm--Liouville difference equation defined by $-q^{-1}D_{-\omega q^{-1},q^{-1}}D_{\omega ,q}y( x) +v(x) y( x) =\lambda y(x)$, $x\in (\omega _{0},\infty),$ where $\lambda$ is a complex parameter, $v$ is a real-valued continuous function at $\omega _{0}$ defined on $[\omega _{0},\infty)$. These type equations are obtained when the ordinary derivative in the classical Sturm--Liouville problem is replaced by the $\omega,q$-Hahn difference operator $D_{\omega,q}$. We develop the $\omega,q$-analogue of the classical Titchmarsh--Weyl theory for such equations. In other words, we study the existence of square-integrable solutions of the singular Hahn--Sturm--Liouville equation. Accordingly, first we define an appropriate Hilbert space in terms of Jackson--N\"{o}rlund integral and then we study families of regular Hahn--Sturm--Liouville problems on $[\omega_{0},q^{-n}]$, $n\in \mathbb{N}$. Then we define a family of circles that converge either to a point or a circle. Thus, we will define the limit-point, limit-circle cases in the Hahn calculus setting by using Titchmarsh's technique.

Gauging scale symmetry and inflation: Weyl versus Palatini gravity

We present a comparative study of inflation in two theories of quadratic gravity with gauged scale symmetry: (1) the original Weyl quadratic gravity and (2) the theory defined by a similar action but in the Palatini approach obtained by replacing the Weyl connection by its Palatini counterpart. These theories have different vectorial non-metricity induced by the gauge field ($$w_\mu $$ w μ ) of this symmetry. Both theories have a novel spontaneous breaking of gauged scale symmetry, in the absence of matter, where the necessary scalar field is not added ad-hoc to this purpose but is of geometric origin and part of the quadratic action. The Einstein-Proca action (of $$w_\mu $$ w μ ), Planck scale and metricity emerge in the broken phase after $$w_\mu $$ w μ acquires mass (Stueckelberg mechanism), then decouples. In the presence of matter ($$\phi _1$$ ϕ 1 ), non-minimally coupled, the scalar potential is similar in both theories up to couplings and field rescaling. For small field values the potential is Higgs-like while for large fields inflation is possible. Due to their $$R^2$$ R 2 term, both theories have a small tensor-to-scalar ratio ($$r\sim 10^{-3}$$ r ∼ 10 - 3 ), larger in Palatini case. For a fixed spectral index $$n_s$$ n s , reducing the non-minimal coupling ($$\xi _1$$ ξ 1 ) increases r which in Weyl theory is bounded from above by that of Starobinsky inflation. For a small enough $$\xi _1\le 10^{-3}$$ ξ 1 ≤ 10 - 3 , unlike the Palatini version, Weyl theory gives a dependence $$r(n_s)$$ r ( n s ) similar to that in Starobinsky inflation, while also protecting r against higher dimensional operators corrections.

Integrability and cosmological solutions in Einstein-æther-Weyl theory

We consider a Lorentz violating scalar field cosmological model given by the modified Einstein-æther theory defined in Weyl integrable geometry. The existence of exact and analytic solutions is investigated for the case of a spatially flat Friedmann–Lemaître–Robertson–Walker background space. We show that the theory admits cosmological solutions of special interests. In addition, we prove that the cosmological field equations admit the Lewis invariant as a second conservation law, which indicates the integrability of the field equations.

Quasinormal modes of Dirac field in the Einstein–Dilaton–Gauss–Bonnet and Einstein–Weyl gravities

Quasinormal modes of Dirac field in the background of a non-Schwarzschild black holes in theories with higher curvature corrections are investigated in this paper. With the help of the semi-analytic WKB approximation and further using of Padé approximants as prescribed in Matyjasek and Opala (Phys Rev D 96(2):024011. arXiv:1704.00361 [gr-qc], 2017) we consider quasinormal modes of a test massless Dirac field in the Einstein–Dilaton–Gauss–Bonnet (EdGB) and Einstein–Weyl (EW) theories. Even though the effective potential for one of the chiralities has a negative gap we show that the Dirac field is stable in both theories. We find the dependence of the modes on the new dimensionless parameter p (related to the coupling constant in each theory) for different values of the angular parameter $$\ell $$ℓ and show that the frequencies tend to linear dependence on p. The allowed deviations of qausinormal modes from their Schwarzschild limit are one order larger for the Einstein–Weyl theory than for the Einstein–Dilaton–Gauss–Bonnet one, achieving the order of tens of percents. In addition, we test the Hod conjecture which suggests the upper bound for the imaginary part of the frequency of the longest lived quasinormal modes by the Hawking temperature multiplied by a factor. We show that in both non-Schwarzschild metrics the Dirac field obeys the above conjecture for the whole range of black-hole parameters.

Titchmarsh–Weyl theory for q-Dirac systems

In this work, we establish Titchmarsh–Weyl theory for singular [Formula: see text]-Dirac systems. Thus, we extend classical Titchmarsh–Weyl theory for Dirac systems to [Formula: see text]-analogue of this system. We show that it does not occur for the limit-circle case for the [Formula: see text]-Dirac system.

Congruence kinematics in conformal gravity

In this paper we calculate the kinematical quantities possessed by Raychaudhuri equations, tocharacterize a congruence of time-like integral curves, according to the vacuum radial solution of Weyl theory of gravity. Also the corresponding flows are plotted for denfinite values of constants.

Affine quantum Schur algebras at roots of unity

Finite dimensional irreducible modules for the affine quantum Schur algebra [Formula: see text] were classified in [B. Deng, J. Du and Q. Fu, A Double Hall Algebra Approach to Affine Quantum Schur–Weyl Theory, London Mathematical Society Lecture Note Series, Vol. 401 (Cambridge University Press, Cambridge, 2012), Chapt. 4] when [Formula: see text] is not a root of unity. We will classify finite-dimensional irreducible modules for affine quantum Schur algebras at roots of unity and generalize [J. A. Green, Polynomial Representations of [Formula: see text] , 2nd edn., with an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker, Lecture Notes in Mathematics, Vol. 830 (Springer-Verlag, Berlin, 2007), (6.5f) and (6.5g)] to the affine case in this paper.