Singular perturbations and nonstandard analysis

Abstract

We study by methods of nonstandard analysis second order differential operators with zero order coefficients which are too singular to be defined by standard functions. In particular we study perturbations of the Laplacian in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R cubed"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{R^3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> given by potentials of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda normal upper Sigma Subscript j Baseline delta left-parenthesis x minus x Subscript j Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Σ<!-- Σ --></mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:mi>δ<!-- δ --></mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mspace width="thinmathspace" /> <mml:mo>−<!-- − --></mml:mo> <mml:mspace width="thinmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\lambda {\Sigma _j}\delta \left ( {x\, - \,{x_j}} \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also study Sturm-Liouville problems with zero order coefficients given by measures and prove that they satisfy the same oscillation theorems as the regular Sturm-Liouville problems.