8 problems

Let $A_1,\dots,A_N$ be bounded non-Hermitian operators with real spectra in a Hilbert space $\mathcal{H}$. For each $k$, let $A_k^*$ denote the adjoint of $A_k$ in $\mathcal{H}$. A…

Imaginary cubic oscillator spectral conjecture. The spectrum of energies is real, discrete and bounded below. Numerical and semiclassical analysis supports this conjecture. The cla…

Dorey–Dunning–Tateo conjecture. The spectrum of ${\cal H}_{N,l}$ is real and positive for $N\ge 2$ and $l$ small. This is a further ${\cal PT}$-symmetric generalization of the Bess…

Let $H^{(BB)}$ be the Bender–Boettcher Hamiltonian … and let $\mathfrak{h}\to H^{(BB)}$ be Dyson's mapping from a conventional Hermitian Hamiltonian to this PT-symmetric Hamiltonia…

Let $V_{BB}(x,N)=-(ix)^N$. For $N=4,8,12$, consider formula (8), choosing only the MxTP out of the three pairs of complex turning points. Reflectivity-zero spectrum conjecture. For…

Non-reality conjecture. The entire discrete spectrum of this potential cannot be real.

Stokes-wedge pairing conjecture. The number of paths that can possibly lead to distinct real solutions is given by the number of pairs of noncontiguous $\mathcal{PT}$-symmetric Sto…

Sparse-pseudometric construction proposal. In place of choosing the sufficiently elementary Krein-space pseudometric $\mathcal{P}$ in advance, it may prove more efficient to start…