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When a $4D$ supersymmetric theory is placed on $S^3 \times \mathbb{R}$, thesupersymmetric algebra is necessarily modified to $su(2|1)$ and we are dealingwith a weak supersymmetric system. For such systems, the excited states of theHamiltonian are not all paired. As a result, the Witten index Tr$\{(-1)^Fe^{-\beta H}\}$ is no longer an integer number, but a $\beta$-dependentfunction.  However, this function stays invariant under deformations of the theory thatkeep the supersymmetry algebra intact. Based on the Hilbert space analysis, wegive a simple general proof of this fact. We then show how this invarianceworks for two simplest weak supersymmetric quantum mechanical systems involvinga real or a complex bosonic degree of freedom.

Witten index for weak supersymmetric systems: invariance under deformations