This work is devoted to discuss some spectral properties and the scattering function of the impulsive operator generated by the Sturm–Liouville equation. We present a different method to investigate the spectral singularities and eigenvalues of the mentioned operator. We also obtain the finiteness of eigenvalues and spectral singularities with finite multiplicities under some certain conditions. Finally, we illustrate our results by a detailed example.

In the present paper, we introduce the notion of Pythagorean fuzzy topological space by motivating from the notion of fuzzy topological space. We define Pythagorean fuzzy continuity of a function defined between Pythagorean fuzzy topological spaces and we characterize this concept. Using the concept of continuity, we also give a method to construct a Pythagorean fuzzy topology on a given non-empty set.

In this work, we are concerned with di¤erence operator of second order with impulsive condition. By the help of a transfer matrix M , we present scattering function of corresponding operator and examine the spectral properties of this impulsive problem.

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