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We study the properties of two-dimensional conformal field theories (CFTs)with sine-square deformation (SSD). We show that there are no eigenstates offinite norm for the Hamiltonian of a unitary CFT with SSD, except for thezero-energy vacuum state $|0\rangle$. We then introduce a regularized versionof the SSD Hamiltonian which is related to the undeformed Hamiltonian via aunitary transformation corresponding to the Mobius quantization. The unitaryequivalence of the two Hamiltonians allows us to obtain zero-energy states ofthe deformed Hamiltonian in a systematic way. The regularization also providesa way to compute the expectation values of observables in zero-energy statesthat are not necessarily normalizable.

Zero-energy states in conformal field theory with sine-square deformation