Related papers: Which Metrics Are Consistent with a Given Pseudo-H…

Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential,…

Associated to an Hadamard matrix $H\in M_N(\mathbb C)$ is the spectral measure $\mu\in\mathcal P[0,N]$ of the corresponding Hopf image algebra, $A=C(G)$ with $G\subset S_N^+$. We study here a certain family of discrete measures…

An array of N subsequent Laguerre polynomials is interpreted as an eigenvector of a non-Hermitian tridiagonal Hamiltonian $H$ with real spectrum or, better said, of an exactly solvable N-site-lattice cryptohermitian Hamiltonian whose…

We consider $n\times n$ non-Hermitian random matrices with independent entries and a variance profile, as well as an additive deterministic diagonal deformation. We show that their empirical eigenvalue distribution converges to a limiting…

For a diagonalizable linear operator $H:\mathscr{H}\to\mathscr{H}$ acting in a separable Hilbert space $\mathscr{H}$, i.e., an operator with a purely point spectrum, eigenvalues with finite algebraic multiplicities, and a set of…

We define pseudo-reality and pseudo-adjointness of a Hamiltonian, $H$, as $\rho H \rho^{-1}=H^\ast$ and $\mu H \mu^{-1}=H^\prime$, respectively. We prove that the former yields the {\it necessary} condition for spectrum to be real whereas…

We study the interplay between the following types of special non-K\"ahler Hermitian metrics on compact complex manifolds: it locally conformally K\"ahler, $k$-Gauduchon, balanced and locally conformally balanced and prove that a locally…

We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the…

Quantum bound-state energies are assumed generated by PT-symmetric Hamiltonians H where P is, typically, parity. It is known that their spectrum only remains real and observable (i.e., in the language of physics, the PT-symmetry remains…

We prove that in finite dimensions, a Parity-Time (PT)-symmetric Hamiltonian is necessarily pseudo-Hermitian regardless of whether it is diagonalizable or not. This result is different from Mostafazadeh's, which requires the Hamiltonian to…

For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ of elements in a unital Banach algebra ${\mathcal B}$, its projective joint spectrum $P(A)$ is the collection of $z\in {\mathbb C}^n$ such that $A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ is not invertible.…

We consider pairs of anti-commuting $2p$-by-$2p$ Hermitian matrices that are chosen randomly with respect to a Gaussian measure. Generically such a pair decomposes into the direct sum of $2$-by-$2$ blocks on which the first matrix has…

The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly…

Unitary 1-matrix models are shown to be exactly equivalent to hermitian 1-matrix models coupled to 2N vectors with appropriate potentials, to all orders in the 1/N expansion. This fact allows us to use all the techniques developed and…

Call an $n$-by-$n$ invertible matrix $S$ a \emph{Perron similarity} if there is a real non-scalar diagonal matrix $D$ such that $S D S^{-1}$ is entrywise nonnegative. We give two characterizations of Perron similarities and study the…

Given two Hermitian matrices, $A$ and $B$, we introduce a new type of spectral measure, a $\textit{tracial joint spectral measure}$ $\mu_{A, B}$ on the plane. Existence of this measure implies the following two results: 1) any…

This paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition $H^\dagger=H$ on the Hamiltonian, where $\dagger$ represents the mathematical operation of complex conjugation and matrix…

In the context of two particularly interesting non-Hermitian models in quantum mechanics we explore the relationship between the original Hamiltonian H and its Hermitian counterpart h, obtained from H by a similarity transformation, as…

The spectrum of the Hermitian Hamiltonian $H=p^2+V(x)$ is real and discrete if the potential $V(x)\to\infty$ as $x\to\pm\infty$. However, if $V(x)$ is complex and PT-symmetric, it is conjectured that, except in rare special cases, $V(x)$…

The adjacency matrices of graphs form a special subset of the set of all integer symmetric matrices. The description of which graphs have all their eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most 2) has been…