On spectral structure and spectral eigenvalue problems for a class of self similar spectral measure with product form

Abstract

Abstract Let μ be a Borel probability measure with compact support on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:math> . We say that μ is a spectral measure if there exists <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:math> , called a spectrum of μ , such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">Λ</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>≔</mml:mo> <mml:msub> <mml:mrow> <mml:mrow> <mml:mo stretchy="false">{</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mi>i</mml:mi> <mml:mrow> <mml:mo stretchy="false">⟨</mml:mo> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy="false">⟩</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">}</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="normal">Λ</mml:mi> </mml:mrow> </mml:msub> </mml:math> forms an orthonormal basis for L 2 ( μ ). In this paper, we study the structure of spectra for a class of self-similar spectral measure μ R , B with product form on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:math> . We first give a partially characterize for E Λ to be a maximal orthogonal family in L 2 ( μ R , B ) by using the notion of maximal tree mapping. Based on this, we give a sufficient condition for a maximal orthogonal family E Λ (which corresponds to a maximal tree mapping) to be an orthonormal basis of L 2 ( μ R , B ). Moreover, we completely settle two types of spectral eigenvalue problems for μ R , B . Precisely, on the first case, for the model spectrum (simplest spectrum) of μ R , B , we characterize all possible real numbers t such that t Λ is also a spectrum of μ R , B . On the other case, we characterize all possible real numbers t such that there exists a countable set Λ such that Λ and t Λ are both spectra of μ R , B .

Topics & Concepts