On the Positive-Definiteness of an Anisotropic Operator

Abstract

We study the positive-definiteness of a family of $L^2(\mathbf{R})$ integral operators with kernel $K_{t,a}(x,y)=(1+(x-y{)}^2+a(x^2+y^2{)}^t{)}^{-1}$, with $t>0$ and $a>0$. When $0<t\le 1$, the known theory of positive-definite kernels ensures that the operator is positive-definite; when $t>1$, constructions disprove positive-definiteness for many $(t,a)$-pairs.

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References in corpus (3)

• Non-Fermi liquid and pairing in electron-doped cuprates
• Theory of non-Fermi liquid and pairing in electron-doped cuprates
• A family of anisotropic integral operators and behaviour of its maximal eigenvalue