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We prove a unique continuation principle or uncertainty relation valid for Schr\"odinger operator eigenfunctions, or more generally solutions of a Schr\"odinger inequality, on cubes of side $L\in 2\NN+1$. It establishes an equi-distribution…
We study the (pseudo-) anyon Hubbard model on a one-dimensional lattice without the presence of a three-body hardcore constraint. In particular, for the pseudo-fermion limit of a large statistical angle $\theta\approx\pi$, we observe a…
The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. We propose an…
The O(4) supersymmetry of the hydrogen atom is utilized to construct a complete basis using only the bound state wave functions. For a large class of perturbations, an expansion of the electron (exciton) wave function into such a complete…
We develop a general approach to study three-dimensional Schroedinger operators with confining potentials depending on the distance to a surface. The main idea is to apply parallel coordinates based on the surface but outside its cut locus…
A model operator $H$ associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The precise location and structure of the essential spectrum of…
We show that for a particular model, the quantum mechanical bootstrap is capable of finding exact results. We consider a solvable system with Hamiltonian $H=SZ(1-Z)S$, where $Z$ and $S$ satisfy canonical commutation relations. While this…
The origin of pseudospin symmetry (PSS) and its breaking mechanism are explored by combining supersymmetry (SUSY) quantum mechanics, perturbation theory, and the similarity renormalization group (SRG) method. The Schr\"odinger equation is…
The state overlap, quantified via $\tr[\rho \sigma]$, is a metric widely used to assess the closeness between two quantum states $\rho$ and $\sigma$. Although global state overlap alone does not directly capture entanglement properties, we…
We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded…
A set $S\subseteq \re^n$ is called to be {\it Semidefinite (SDP)} representable if $S$ equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). The contributions of this paper…
We establish $\frac{1}{2}$-H\"older continuity, or even the Lipschitz property, for the spectral measures of half-line discrete Schr\"odinger operators under suitable boundary conditions and exponentially decaying small potentials. These…
Quantum parameter estimation theory is an important component of quantum information theory and provides the statistical foundation that underpins important topics such as quantum system identification and quantum waveform estimation. When…
Supersymmetric quantum mechanics is well known to provide, together with the so-called shape invariance condition, an elegant method to solve the eigenvalue problem of some one-dimensional potentials by simple algebraic manipulations. In…
We investigate the eigenstate thermalization properties of the spin-1/2 $XXZ$ model in two-dimensional rectangular lattices of size $L_1\times L_2$ under periodic boundary conditions. Exploiting the symmetry property, we can perform an…
In ${\cal PT}-$symmetric quantum mechanics one of the most characteristic mathematical features of the formalism is the explicit Hamiltonian-dependence of the physical Hilbert space of states ${\cal H}={\cal H}(H)$. Some of the most…
In this paper we consider the simplified form that a recently introduced general operator description of the Hubbard model on the square lattice with $N_a^2\gg 1$ sites, effective transfer integral $t$, and onsite repulsion $U$ has in a…
We study the low-energy properties of a truncated Haldane pseudopotential with maximal half filling, which describes a strongly correlated system of spinless bosons in a cylinder geometry. For this Hamiltonian with either open or periodic…
Spin Hamiltonians, like the Heisenberg model, are used to describe magnetic properties of exchange-coupled molecules and solids. For finite clusters, physical quantities such as heat capacities, magnetic susceptibilities or…
The absolutely continuous spectrum of one-dimensional Schr\"odinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectrum of free and periodic…