45,546 research outputs found

    Degenerate elliptic operators: capacity, flux and separation

    Full text link
    Let S={St}t0S=\{S_t\}_{t\geq0} be the semigroup generated on L_2(\Ri^d) by a self-adjoint, second-order, divergence-form, elliptic operator HH with Lipschitz continuous coefficients. Further let Ω\Omega be an open subset of \Ri^d with Lipschitz continuous boundary Ω\partial\Omega. We prove that SS leaves L2(Ω)L_2(\Omega) invariant if, and only if, the capacity of the boundary with respect to HH is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result.Comment: 18 page

    Crystallization of random matrix orbits

    Full text link
    Three operations on eigenvalues of real/complex/quaternion (corresponding to β=1,2,4\beta=1,2,4) matrices, obtained from cutting out principal corners, adding, and multiplying matrices can be extrapolated to general values of β>0\beta>0 through associated special functions. We show that β\beta\to\infty limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general β\beta self-adjoint matrix with fixed eigenvalues is known as β\beta-corners process. We show that as β\beta\to\infty these eigenvalues crystallize on the irregular lattice of all the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field (dGFF) put on top of this lattice, which provides a new explanation of why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.Comment: 25 pages. v2: misprints corrected, to appear in IMR

    A rectangular additive convolution for polynomials

    Full text link
    We define the rectangular additive convolution of polynomials with nonnegative real roots as a generalization of the asymmetric additive convolution introduced by Marcus, Spielman and Srivastava. We then prove a sliding bound on the largest root of this convolution. The main tool used in the analysis is a differential operator derived from the "rectangular Cauchy transform" introduced by Benaych-Georges. The proof is inductive, with the base case requiring a new nonasymptotic bound on the Cauchy transform of Gegenbauer polynomials which may be of independent interest

    Degenerate elliptic operators in one dimension

    Full text link
    Let HH be the symmetric second-order differential operator on L_2(\Ri) with domain C_c^\infty(\Ri) and action Hφ=(cφ)H\varphi=-(c \varphi')' where c\in W^{1,2}_{\rm loc}(\Ri) is a real function which is strictly positive on \Ri\backslash\{0\} but with c(0)=0c(0)=0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of HH. In particular if ν=ν+ν\nu=\nu_+\vee\nu_- where ν±(x)=±±x±1c1\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1} then HH has a unique self-adjoint extension if and only if ν∉L2(0,1)\nu\not\in L_2(0,1) and a unique submarkovian extension if and only if ν∉L(0,1)\nu\not\in L_\infty(0,1). In both cases the corresponding semigroup leaves L2(0,)L_2(0,\infty) and L2(,0)L_2(-\infty,0) invariant. In addition we prove that for a general non-negative c\in W^{1,\infty}_{\rm loc}(\Ri) the corresponding operator HH has a unique submarkovian extension.Comment: 28 page