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Continuous Matrix Product States (cMPS) are powerful variational ansatzstates for ground states of continuous quantum field theories in (1+1)dimension. In this paper we introduce a novel parametrization of the cMPS wavefunction based on basis-spline functions, which we coin spline-based MPS(spMPS), and develop novel regauging techniques for inhomogeneous cMPS. Weextend a recently developed ground-state optimization algorithm fortranslational invariant cMPS [M. Ganahl, J. Rinc\'on, G.Vidal. Phys.Rev.Lett.118,220402 (2017)] to the case of inhomogeneous cMPS and, asproof-of-principle, use it to obtain the ground-state of a gas of Lieb-Linigerbosons in a periodic potential. The proposed method provides a first workingimplementation of a cMPS optimization for non-translational invariantcontinuous Hamiltonians.

Continuous Matrix Product States for Inhomogeneous Quantum Field Theories: a Basis-Spline Approach