The sign of the Green function of an n-th order linear boundary value problem

[EN] This paper provides results on the sign of the Green function (and its partial derivatives) of ann-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions. The dependence of the absolute value of the Green function and some of its partial derivatives with respect to the extremes where the boundary conditions are set is also assessed.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Almenar, P.; Jódar Sánchez, LA. (2020). The sign of the Green function of an n-th order linear boundary value problem. Mathematics. 8(5):1-22. https://doi.org/10.3390/math8050673S12285Butler, G. ., & Erbe, L. . (1983). Integral comparison theorems and extremal points for linear differential equations. Journal of Differential Equations, 47(2), 214-226. doi:10.1016/0022-0396(83)90034-7Peterson, A. C. (1979). Green’s functions for focal type boundary value problems. Rocky Mountain Journal of Mathematics, 9(4). doi:10.1216/rmj-1979-9-4-721Peterson, A. C. (1980). Focal Green’s functions for fourth-order differential equations. Journal of Mathematical Analysis and Applications, 75(2), 602-610. doi:10.1016/0022-247x(80)90104-3Elias, U. (1980). Green’s functions for a non-disconjugate differential operator. Journal of Differential Equations, 37(3), 318-350. doi:10.1016/0022-0396(80)90103-5Nehari, Z. (1967). Disconjugate linear differential operators. Transactions of the American Mathematical Society, 129(3), 500-500. doi:10.1090/s0002-9947-1967-0219781-0Keener, M. S., & Travis, C. C. (1978). Positive Cones and Focal Points for a Class of nth Order Differential Equations. Transactions of the American Mathematical Society, 237, 331. doi:10.2307/1997625Schmitt, K., & Smith, H. L. (1978). Positive solutions and conjugate points for systems of differential equations. Nonlinear Analysis: Theory, Methods & Applications, 2(1), 93-105. doi:10.1016/0362-546x(78)90045-7Eloe, P. W., Hankerson, D., & Henderson, J. (1992). Positive solutions and conjugate points for multipoint boundary value problems. Journal of Differential Equations, 95(1), 20-32. doi:10.1016/0022-0396(92)90041-kEloe, P. W., & Henderson, J. (1994). Focal Point Characterizations and Comparisons for Right Focal Differential Operators. Journal of Mathematical Analysis and Applications, 181(1), 22-34. doi:10.1006/jmaa.1994.1003Almenar, P., & Jódar, L. (2015). Solvability ofNth Order Linear Boundary Value Problems. International Journal of Differential Equations, 2015, 1-19. doi:10.1155/2015/230405Almenar, P., & Jódar, L. (2016). Improving Results on Solvability of a Class ofnth-Order Linear Boundary Value Problems. International Journal of Differential Equations, 2016, 1-10. doi:10.1155/2016/3750530Almenar, P., & Jodar, L. (2017). SOLVABILITY OF A CLASS OF N -TH ORDER LINEAR FOCAL PROBLEMS. Mathematical Modelling and Analysis, 22(4), 528-547. doi:10.3846/13926292.2017.1329757Sun, Y., Sun, Q., & Zhang, X. (2014). Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem. Abstract and Applied Analysis, 2014, 1-7. doi:10.1155/2014/513051Hao, X., Liu, L., & Wu, Y. (2015). Iterative solution to singular nth-order nonlocal boundary value problems. Boundary Value Problems, 2015(1). doi:10.1186/s13661-015-0393-6Webb, J. R. L. (2017). New fixed point index results and nonlinear boundary value problems. Bulletin of the London Mathematical Society, 49(3), 534-547. doi:10.1112/blms.12055Jiang, D., & Yuan, C. (2010). The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Analysis: Theory, Methods & Applications, 72(2), 710-719. doi:10.1016/j.na.2009.07.012Wang, Y., & Liu, L. (2017). Positive properties of the Green function for two-term fractional differential equations and its application. The Journal of Nonlinear Sciences and Applications, 10(04), 2094-2102. doi:10.22436/jnsa.010.04.63Zhang, L., & Tian, H. (2017). Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations. Advances in Difference Equations, 2017(1). doi:10.1186/s13662-017-1157-7Wang, Y. (2020). The Green’s function of a class of two-term fractional differential equation boundary value problem and its applications. Advances in Difference Equations, 2020(1). doi:10.1186/s13662-020-02549-

Hawking Spectrum and High Frequency Dispersion

We study the spectrum of created particles in two-dimensional black hole geometries for a linear, hermitian scalar field satisfying a Lorentz non-invariant field equation with higher spatial derivative terms that are suppressed by powers of a fundamental momentum scale $k_0$. The preferred frame is the ``free-fall frame" of the black hole. This model is a variation of Unruh's sonic black hole analogy. We find that there are two qualitatively different types of particle production in this model: a thermal Hawking flux generated by ``mode conversion" at the black hole horizon, and a non-thermal spectrum generated via scattering off the background into negative free-fall frequency modes. This second process has nothing to do with black holes and does not occur for the ordinary wave equation because such modes do not propagate outside the horizon with positive Killing frequency. The horizon component of the radiation is astonishingly close to a perfect thermal spectrum: for the smoothest metric studied, with Hawking temperature $T_H\simeq0.0008k_0$, agreement is of order $(T_H/k_0)^3$ at frequency $\omega=T_H$, and agreement to order $T_H/k_0$ persists out to $\omega/T_H\simeq 45$ where the thermal number flux is $O(10^{-20}$). The flux from scattering dominates at large $\omega$ and becomes many orders of magnitude larger than the horizon component for metrics with a ``kink", i.e. a region of high curvature localized on a static worldline outside the horizon. This non-thermal flux amounts to roughly 10\% of the total luminosity for the kinkier metrics considered. The flux exhibits oscillations as a function of frequency which can be explained by interference between the various contributions to the flux.Comment: 32 pages, plain latex, 16 figures included using psfi

Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

We develop techniques to compute multi-instanton corrections to the 1/N expansion in matrix models described by orthogonal polynomials. These techniques are based on finding trans-series solutions, i.e. formal solutions with exponentially small corrections, to the recursion relations characterizing the free energy. We illustrate this method in the Hermitian, quartic matrix model, and we provide a detailed description of the instanton corrections in the Gross-Witten-Wadia (GWW) unitary matrix model. Moreover, we use Borel resummation techniques and results from the theory of resurgent functions to relate the formal multi-instanton series to the nonperturbative definition of the matrix model. We study this relation in the case of the GWW model and its double-scaling limit, providing in this way a nice illustration of various mechanisms connecting the resummation of perturbative series to nonperturbative results, like the cancellation of nonperturbative ambiguities. Finally, we argue that trans-series solutions are also relevant in the context of topological string theory. In particular, we point out that in topological string models with both a matrix model and a large N gauge theory description, the nonperturbative, holographic definition involves a sum over the multi-instanton sectors of the matrix modelComment: 50 pages, 12 figures, comments and references added, small correction

On non-uniform smeared black branes

We investigate charged dilatonic black $p$-branes smeared on a transverse circle. The system can be reduced to neutral vacuum black branes, and we perform static perturbations for the reduced system to construct non-uniform solutions. At each order a single master equation is derived, and the Gregory-Laflamme critical wavelength is determined. Based on the non-uniform solutions, we discuss thermodynamic properties of this system and argue that in a microcanonical ensemble the non-uniform smeared branes are entropically disfavored even near the extremality, if the spacetime dimension is $D \le 13 +p$, which is the critical dimension for the vacuum case. However, the critical dimension is not universal. In a canonical ensemble the vacuum non-uniform black branes are thermodynamically favorable at $D > 12+p$, whereas the non-uniform smeared branes are favorable at $D > 14+p$ near the extremality.Comment: 24 pages, 2 figures; v2: typos corrected, submitted to Class.Quant.Gra