  We study the phenomenon of spontaneous symmetry breaking in dissipationless
resonant tunneling heterostructures (RTS). To describe the quantum transport in
this system we apply both the nonequilibrium Green function formalism based on
a tight-binding model and a numerical solution of the Schroedinger equation
within the envelope wavefunction formalism. An auxiliary non-Hermitian
Hamiltonian is introduced. Its eigenvalues determine exactly the transparency
peak positions. We present a procedure how to construct a family of
non-Hermitian Hamiltonians with real eigenvalues. In general these Hamiltonians
do not have PT-symmetry. In spatially symmetric RTS the corresponding auxiliary
non-Hermitian Hamiltonian becomes PT-symmetric and possesses real eigenvalues,
which can coalesce at exceptional points (EP) of Hamiltonian. A coalescence of
the auxiliary non-Hermitian Hamiltonian eigenvalues means a coalescence of
resonances in RTS, which is accompanied be symmetry breaking of the electron
wavefunction probability distribution (at a given direction of the particle
flow). We construct a classification of different types of the peak coalescence
in terms of the catastrophe theory and investigate the impact of dissipation
and asymmetry on these phenomena. Possible applications include sensors and
broad-band filters.
