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Abstract Integrals of products of Bessel functions exhibit an intriguing feature: they may vanish identically, not due to an orthogonality property, but rather when certain conditions on the parameters specifying the integrand are satisfied. We provide a physical interpretation of this feature in the context of both single-particle and many-body properties of electrons on a lattice (“Bloch electrons“), namely, in terms of their density of states and umklapp scattering rate. (In an umklapp event, the change in the momentum of two colliding electrons is equal to a reciprocal lattice vector, which gives rise to a finite resistivity due to electron-electron interaction.) In this context, the vanishing of an integral follows simply from the condition that either the density of states vanishes due to the electron energy lying outside the band in which free propagation of electron waves is allowed, or that an umklapp process is kinematically forbidden due to the Fermi surface being smaller than a critical value. 

Abstract The longstanding view of the zero sound mode in a Fermi liquid is that for repulsive interaction it resides outside the particle-hole continuum and gives rise to a sharp peak in the corresponding susceptibility, while for attractive interaction it is a resonance inside the particle-hole continuum. We argue that in a two-dimensional Fermi liquid there exist two additional types of zero sound: “hidden” and “mirage” modes. A hidden mode resides outside the particle-hole continuum already for attractive interaction. It does not appear as a sharp peak in the susceptibility, but determines the long-time transient response of a Fermi liquid and can be identified in pump-probe experiments. A mirage mode emerges for strong enough repulsion. Unlike the conventional zero sound, it does not correspond to a true pole, yet it gives rise to a peak in the particle-hole susceptibility. It can be detected by measuring the width of the peak, which for a mirage mode is larger than the single-particle scattering rate.