This study provides a comprehensive analysis of $S$-wave exotic hydrogen-like three-body systems $(pp{\mu }^{-},$ $pp{\tau }^{-},$ ${\mu }^{-}{\mu }^{-}p,$ ${\tau }^{-}{\tau }^{-}p,$ $p{\mu }^{-}{\tau }^{-})$ with spin parity $J^P=1/2^{+}$ and $3/2^{+}$, and four-body systems $(pp{\mu }^{-}{\mu }^{-},$ $pp{\tau }^{-}{\tau }^{-})$ with $J^P=0^{+}$, $1^{+}$, and $2^{+}$. The wave functions are constructed using complete spin configurations and spatial components with various Jacobi coordinate structures. We use complex scaling and Gaussian expansion methods to solve the complex-scaled Schrödinger equation and obtain possible bound and quasibound states. The resulting binding energies range from $-33.8$ keV to $-340\mathrm{eV}$, with the $p{\mu }^{-}{\tau }^{-}$ system being the only case where no bound state is found. Furthermore, we present theoretical estimation of the bound-state energy levels of $pp{\mu }^{-}{\mu }^{-}$ and $pp{\tau }^{-}{\tau }^{-}$, which is of significant importance for understanding exotic few-body Coulomb systems. We also analyze spin configurations and root-mean-square radii to elucidate the spatial structure of these bound and quasibound states. Our results reveal that $K$-type spatial configurations play a crucial role in accurately describing bound and quasibound states in the hydrogen-molecule-like systems $pp{\mu }^{-}{\mu }^{-}$ and $pp{\tau }^{-}{\tau }^{-}$. Incorporating $K$-type configurations significantly alters the mass spectra of these states. Future muon colliders and muon facilities may offer promising platforms for the possible copious production of such heavy flavored hydrogen molecules and molecular ions. For instance, scattering processes such as $2{\mu }^{-}+{\mathrm{H}}_2\to {\mathrm{H}}_{2\mu }+2e^{-}$, ${\mu }^{-}+{\mathrm{H}}_2\to {\mathrm{H}}_{\mu \mathrm{e}}+e^{-}$, and ${\mu }^{-}+{\mathrm{H}}_2^{+}\to {\mathrm{H}}_{2\mu }^{+}+e^{-}$ could be utilized, facilitating detailed studies of intriguing states such as ${\mathrm{H}}_{2\mu }$, ${\mathrm{H}}_{\mu \mathrm{e}}$, and ${\mathrm{H}}_{2\mu }^{+}$.

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• Atomic spectra
• Interatomic & molecular potentials
• Nuclear & electron resonance
• Exotic atoms & molecules
• Muonic atoms & molecules
• Lifetimes & widths
• Mass
• Parity
• Spin
• Few-body systems
• Schroedinger equation