58 problems

Let … be the Schrödinger operator with potential $u(x)=ix^3$. Bessis–Zinn-Justin conjecture. The spectrum of $H$ is real. This conjecture concerns the spectral reality of a non-sel…

Let $H$ be a PT-invariant Hamiltonian, and suppose that PT symmetry is unbroken, meaning that the energy eigenstates of $H$ are also eigenstates of the operator $PT$. Reality conje…

Consider the weakly non-Hermitian square-well Hamiltonian with three discontinuities at $x=-l$, $0$, and $l$, whose eigenfunctions and partner Hamiltonians are invariant under comb…

A $CPT$-symmetric field theory is a field theory invariant under the combined action of charge conjugation, parity, and time reversal. A field theory is non-Hermitian when its Hami…

Let a Schrödinger Hamiltonian be invariant under the joint action of parity (P) and time reversal (T) transformations. Bender's conjecture. Under this less restrictive condition th…

Let … where $g$ is real, so that the cubic oscillator has a purely imaginary coupling constant. Bessis's reality conjecture. All eigenvalues of $H$ are real. The conjecture concern…

Bessis–Zinn-Justin and Bender–Boettcher conjectures. In the first case, all eigenvalues are real and positive; in the second case, all eigenvalues are real and positive.

Exponential pulling-through conjecture. The only complex eigenvalues of this problem should be those arising when an exponential term allows the quantisation-condition function to…

Let ${\cal PT}$-symmetric quantum mechanics denote the class of non-Hermitian quantum-mechanical systems invariant under combined parity and time reversal, and let pseudo-Hermitian…

For $obreaktextstyledelta0$ and $g0$, consider the power-law Hamiltonians … The special cases $obreaktextstyledelta=2$ and $obreaktextstyledelta=1$ reproduce the quartic and cub…

In quantum mechanics, let $H$ be a Hamiltonian with $H\neq H^\dagger$, where $H^\dagger$ denotes the Hermitian adjoint, and let $H^\ddagger$ denote the weaker analogue of Hermitici…

Let a quantum system be invariant under combined parity and time reversal, denoted ${\cal P}{\cal T}$, and let its discrete eigenenergies be $E$. Bender–Boettcher reality conjectur…

A Hamiltonian is assumed to possess ${\cal PCT}$ symmetry, and this symmetry is said to be spontaneously broken when the Hamiltonian's eigenstates do not respect it. Reality-and-po…

Consider the eigenvalue problem … where $H=-\frac{d^2}{dx^2}-(ix)^3$ acts on the real line. Bessis's conjecture. All eigenvalues of $H$ are real and positive. This is a conjecture…

Consider the complex eigenfunctions associated with a $\mathcal{PT}$-symmetric Hamiltonian, and scale their zeros by fixing the magnitudes of the turning points relative to a unit…

Reality conjecture. The reality of the eigenvalues $\lambda_k$ should hold for all real $\alpha$ and $l$.

Bessis–Zinn-Justin–Bender–Boettcher conjecture. The eigenvalues $\lambda_k$ are real and positive for $M\geq 1$.

The $1+1$-dimensional nonlinear Dirac equation admits $mathcal{PT}$-symmetric perturbations, including the Alexeeva-Barashenkov-Saxena (ABS) model. A conservation law is a quanti…

PT-inner-product conjecture. The appropriate inner product is conjectured to be

PT-symmetry real-spectrum conjecture. The eigenvalues of any ${\mathcal{PT}}$-symmetric Hamiltonian should all be real.

Universality conjecture. These observed features occur universally in the spontaneously broken ${\cal PT}$-regimes of non-Hermitian systems: the signs of the heat supply or removal…

The scattering problem concerns finite, truncated potentials with reflectionless above-barrier states that propagate as plane waves at infinity. The corresponding unbounded potenti…

Consider a dynamical system with balanced loss and gain, whose equations of motion have no velocity-mediated interaction. Decompose its equations into linear and nonlinear parts, a…

Let $\varepsilon$ be the parameter in the classical Hamiltonian, and consider gaps on the negative-imaginary axis of the principal sheet of the Riemann surface of complex trajector…

Let $R_n$ denote regions of the complex classical trajectory Riemann surface, let $t_n$ be pairs of turning points, let $s_n$ be terminating trajectories joining the pair $t_n$, an…