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We consider Bethe subalgebras B(C) in the Yangian $Y(\mathfrak{gl}_2)$ with$C$ regular $2\times 2$ matrix. We study the action of Bethe subalgebras of$Y(\mathfrak{gl}_2)$ on finite-dimensional representations of$Y(\mathfrak{gl}_2)$. We prove that $B(C)$ with real diagonal $C$ has simplespectrum on any irreducible $Y(\mathfrak{gl}_2)$-module corresponding to adisjoint union of real strings. We extend this result to limits of Bethealgebras. Our main tool is the computation of Shapovalov-type determinant forthe nilpotent degeneration of $B(C)$.

Simplicity of spectra for Bethe subalgebras in $Y(\mathfrak{gl}_2)$