Coupling of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>X</mml:mi><mml:mspace width="0.2em"/><mml:mmultiscripts><mml:mi mathvariant="bold-italic">Σ</mml:mi><mml:none/><mml:mo>+</mml:mo><mml:mprescripts/><mml:none/><mml:mn>1</mml:mn></mml:mmultiscripts></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>a</mml:mi><mml:mspace width="0.2em"/><mml:mmultiscripts><mml:mi mathvariant="bold">Σ</mml:mi><mml:none/><mml:mo>+</mml:mo><mml:mprescripts/><mml:none/><mml:mn>3</mml:mn></mml:mmultiscripts></mml:mrow></mml:math>states of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>KRb</mml:mi></mml:mrow></mml:math>

A comprehensive study of the electronic states at the $4s+5s$ asymptote in $\mathrm{KRb}$ is presented. Abundant spectroscopic data on the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ state were collected by Fourier-transform spectroscopy, which allows one to determine an accurate experimental potential energy curve up to $14.8\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$. The existing data set [C. Amiot et al., J. Chem. Phys. 112, 7068 (2000)] on the ground state $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ was extended by several additional levels lying close to the atomic asymptote. In a coupled channels fitting routine complete molecular potentials for both electronic states were fitted. Along with the line frequencies of the molecular transitions, recently published positions of Feshbach resonances in $^{40}\mathrm{K}$ and $^{87}\mathrm{Rb}$ mixtures [F. Ferlaino et al., Phys. Rev. A 74, 039903 (2006)] were included in the fit. This makes the derived potential curves capable for an accurate description of observed cold collision features so far. Predictions of scattering lengths and Feshbach resonances in other isotopic combinations are reported.