22 problems

Bender–Boettcher conjecture. There might exist certain manifestly non-Hermitian realizations of quantum systems that still admit the conventional probabilistic interpretation and p…

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…

The paper discusses coupled Bose–Einstein systems with dissipative effects, described by effective Hamiltonians that need not be Hermitian. Simple effective-Hamiltonian conjecture.…

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 … be the Hamiltonian obtained from $H_\alpha=P^2+iX^3+i\alpha X$ by setting $\alpha=0$, and consider its $L^2$ eigenstates and corresponding eigenenergies. Bessis' conjecture.…

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 quantum-mechanical Hamiltonian is traditionally taken to be Hermitian, meaning that it equals its adjoint, and its spectrum is the set of its energy eigenvalues. Hermiticity conj…

Bender–Boettcher conjecture. The ubiquitous requirement that observables be self-adjoint might be over-restrictive.

The discussion concerns a non-Hermitian Hamiltonian with real spectrum that governs unitary evolution after the Hilbert space of states is appropriately amended. Bender–Boettcher c…

In a non-Hermitian but hermitizable quantum system, let $H$ be a Hamiltonian equipped with a metric operator $\Theta$ defining the inner product…

Let $H=H(\lambda)\neq H^\dagger(\lambda)$ be a one-parameter family of non-Hermitian quantum Hamiltonians satisfying … Let ${\cal D}_{(physical)}=(0,\infty)$ be the admissible open…

Let $H$ be a non-Hermitian Hamiltonian, and let its spectrum denote the usual spectral data while its pseudospectrum denotes the corresponding numerical construction used to assess…

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…

The time-dependent quasi-Hermiticity and Dyson relations are considered for non-Hermitian higher spin models, with the Dyson map components determined by decoupled equations and th…

Amore et al.'s conjecture. Of all the examples of antiunitary symmetry studied so far, PT symmetry appears to be the less likely to be broken.

Let $R_{left}(k)$ and $R_{right}(k)$ denote the reflection probabilities for incidence from the left and right, respectively, and let $T(k)$ denote the transmission probability at…

PT-symmetry spectral conjecture. The positivity and discreteness of the spectrum of $H$ should follow from its invariance under the combined space-reflection and time-reversal oper…

Let $S$ be the spin, let $\gamma$ be the deformation parameter, and let $\boldsymbol{H}$ be the Hamiltonian of a spin chain with inhomogeneous coupling constants $a_n$. Positive-co…

Let $N\geq 3$, let $S$ be the spin, and let $a\neq 0$. An alternating chain has couplings $a,-a,a,-a,\ldots$, and $\boldsymbol{H}^{S,0}_{\{a,-a,a,-a,\ldots\}}$ denotes its spin-cha…

Let $S$ and $s$ be the spin parameters, let $a_1,a_2$ be coupling constants with $a_1a_2<0$, and let $\boldsymbol{H}^{S,s}_{\{a_1,a_2\}}$ be the corresponding two-coupling spin-cha…

Let $H=p^2+V(x)$ be a non-Dirac-Hermitian Hamiltonian with a complex, ${\cal PT}$-symmetric potential $V(x)$. A potential is analytic if it is analytic as a function of $x$, and th…