Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators

Abstract This paper investigates uniqueness results for perturbed periodic Schrödinger operators on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> . Specifically, we consider operators of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo>+</mml:mo> <mml:mi>v</mml:mi> </mml:mrow> </mml:math> , where Δ is the discrete Laplacian, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy="false">→</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> </mml:math> is a periodic potential, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy="false">→</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> </mml:math> represents a decaying impurity. We establish quantitative conditions under which the equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>v</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> </mml:math> , for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> </mml:math> , admits only the trivial solution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes of λ .