We find that a gauged matrix model of rectangular fermionic matrices (a matrix version of the fermion harmonic oscillator) realizes a quantum hall droplet with manifest particle-hole symmetry. The droplet consists of free fermions on the topology of a sphere. It is also possible to deform the Hamiltonian by double trace operators, and we argue that this device can produce two body potentials wh...

We consider Hermitian and symmetric random band matrices H in d > 1 dimensions. The matrix elements Hxy, indexed by x, y ∈ Λ ⊂ Z, are independent, uniformly distributed random variables if |x − y| is less than the band width W , and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales t W . We also show that the localiza...

We discuss a method called quasi-sparse eigenvector diagonalization which finds the most important basis vectors of the low energy eigenstates of a quantum Hamiltonian. It can operate using any basis, either orthogonal or non-orthogonal, and any sparse Hamiltonian, either Hermitian, non-Hermitian, finite-dimensional, or infinite-dimensional. The method is part of a new computational approach wh...

The distribution of the eigenvalues of a Hermitian matrix (or of a Hermitian matrix pencil) reveals important features of the underlying problem, whether a Hamiltonian system in physics, or a social network in behavioral sciences. However, computing all the eigenvalues explicitly is prohibitively expensive for real-world applications. This paper presents two types of methods to efficiently esti...

The quantum-mechanical framework in which observables are associated with Hermitian operators is too narrow to discuss measurements of such important physical quantities as elapsed time or harmonic-oscillator phase. We introduce a broader framework that allows us to derive quantum-mechanical limits on the precision to which a parameter e.g., elapsed time may be determined via arbitrary data ana...

We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature is (non-Hermitian) system matrix has a positive definite semidefinite Hermitian part. In can solve iteratively by Krylov subspace methods based...

We consider the (2+1)-dimensional gauged Heisenberg ferromagnet model coupled with the Chern-Simons gauge fields. Self-dual Chern-Simons solitons, the static zero energy solution saturating Bogomol’nyi bounds, are shown to exist when the generalized spin variable is valued in the Hermitian symmetric spaces G/H . By gauging the maximal torus subgroup of H , we obtain self-dual solitons which sat...

A one-to-one correspondence is known to exist between the spectra of the discrete states of the non Hermitian Swanson-type Hamiltonian H = A†A+αA2+βA† 2, (α 6= β), and an equivalent Hermitian Schrödinger Hamiltonian h, the two Hamiltonians being related through a similarity transformation. In this work we consider the continuum states of h, and examine the nature of the corresponding states of ...

New types of irreducible second order Darboux transformations for the one dimensional Schrödinger equation are described. The main feature of such transformations is that the transformation functions have the eigenvalues grater then the ground state energy of the initial (or reference) Hamiltonian. When such a transformation is presented as a chain of two first order transformations, an interme...