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Singularities of the joint density of states (JDOS) and Fourier-transformedlocal density of states (FT-LDOS) correspond to the hot spots in quasiparticleinterference patterns. In this paper the singularity of JDOS is analyzedexactly, with three types of singularities being classified. In particular, thethird type of singularities are found exactly to be envelopes of the contoursof constant energy. Remarkably, we show that JDOS and FT-LDOS have the samesingular points. Approaching to the singular points, both quantities divergecomplementarily in an inverse-square-root manner if the joint curvature isnonzero. The relative magnitude of divergence is governed by the jointcurvature as well as the product of the quasiparticle velocities. If the jointcurvature of certain singularity is zero, the divergence has a higher orderthan -1/2.

Exact analysis on the singularity of the joint density of states and its relationship to the quasiparticle interference