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The formation of a deeply-bound $K^- pp$ state with $I=1/2$, $J^\pi=0^-$ bythe $^3$He(in-flight $K^-$, $n$) reaction is theoretically investigated in adistorted-wave impulse approximation using the Green's function method. Theexpected inclusive and semi-exclusive spectra at $p_{K^-} = 1.0$ GeV/c and$\theta_{\rm lab} = 0^{\circ}$ are calculated for the forthcoming J-PARC E15experiment. We demonstrate these spectra with several types of phenomenological$K^-$-``$pp$'' optical potentials $U^{\rm opt}(E)$ which have anenergy-dependent imaginary part multiplied by a phase space suppression factor,fitting to recent theoretical predictions or experimental candidates of the$K^-pp$ bound state. The results show that a cusp-like peak at the $\pi \SigmaN$ threshold is an unique signal for the $K^-pp$ bound state in the spectrumincluding the [$K^-pp$] $\to$ $Y + N$ decay process from the two-nucleon $K^-$absorption, as well as a distinct peak of the $K^-pp$ bound state. The shape ofthe spectrum is explained by a trajectory of a moving pole of the $K^-pp$ boundstate in the complex energy plane. The importance of the spectrum with[$K^-pp$] $\to$ $Y + N$ from the two-nucleon $K^-$ absorption is emphasized inorder to extract clear evidence of the $K^-pp$ bound state.

Deeply-bound $K^- pp$ state in the $^3$He(in-flight $K^-$, $n$) spectrum and its moving pole near the $\pi \Sigma N$ threshold