  We consider a superconducting half-wavelength resonator that is grounded at
its both ends and contains a single Josephson junction. Previously this circuit
was considered as a unimon qubit in the single-mode approximation where
dc-phase-biasing the junction to $\pi$ leads to increased anharmonicity and
99.9% experimentally observed single-qubit gate fidelity. Inspired by the
promising first experimental results, we develop here a theoretical and
numerical model for the detailed understanding of the multimode physics of the
unimon circuit. To this end, first, we consider the high-frequency modes of the
unimon circuit and find that even though these modes are at their ground state,
they imply a significant renormalization to the Josephson energy. We introduce
an efficient method how the relevant modes can be fully taken into account and
show that unexcited high-lying modes lead to corrections in the qubit energy
and anharmonicity. Interestingly, provided that the junction is offset from the
middle of the circuit, we find strong cross-Kerr coupling strengths between a
few low-lying modes. This observation paves the way for the utilization of the
multimode structure, for example, as several qubits embedded into a single
unimon circuit.
