Strongly Interacting Quantum Gases in One-Dimensional Traps

Under the second-order degenerate perturbation theory, we show that the physics of $N$ particles with arbitrary spin confined in a one dimensional trap in the strongly interacting regime can be described by super-exchange interaction. An effective spin-chain Hamiltonian (non-translational-invariant Sutherland model) can be constructed from this procedure. For spin-1/2 particles, this model reduces to the non-translational-invariant Heisenberg model, where a transition between Heisenberg anti-ferromagnetic (AFM) and ferromagnetic (FM) states is expected to occur when the interaction strength is tuned from the strongly repulsive to the strongly attractive limit. We show that the FM and the AFM states can be distinguished in two different methods: the first is based on their distinct response to a spin-dependent magnetic gradient, and the second is based on their distinct momentum distribution. We confirm the validity of the spin-chain model by comparison with results obtained from several unbiased techniquesComment: 14 pages, 7 figure

The Commutant of Multiplication by z on the Closure of Rational Functions in Lt(μ)L^t(\mu)

For a compact set $K\subset \mathbb C,$ a finite positive Borel measure $\mu$ on $K,$ and 1 \le t < \i, let $\text{Rat}(K)$ be the set of rational functions with poles off $K$ and let $R^t(K, \mu)$ be the closure of $\text{Rat}(K)$ in $L^t(\mu).$ For a bounded Borel subset $\mathcal D\subset \mathbb C,$ let \area_{\mathcal D} denote the area (Lebesgue) measure restricted to $\mathcal D$ and let H^\i (\mathcal D) be the weak-star closed sub-algebra of L^\i(\area_{\mathcal D}) spanned by $f,$ bounded and analytic on $\mathbb C\setminus E_f$ for some compact subset $E_f \subset \mathbb C\setminus \mathcal D.$ We show that if $R^t(K, \mu)$ contains no non-trivial direct $L^t$ summands, then there exists a Borel subset $\mathcal R \subset K$ whose closure contains the support of $\mu$ and there exists an isometric isomorphism and a weak-star homeomorphism $\rho$ from $R^t(K, \mu) \cap L^\infty(\mu)$ onto $H^\infty(\mathcal R)$ such that $\rho(r) = r$ for all $r\in\text{Rat}(K).$ Consequently, we obtain some structural decomposition theorems for \rtkmu.Comment: arXiv admin note: text overlap with arXiv:2212.1081

Multiplication Operators on Hilbert Spaces

Let $S$ be a subnormal operator on a separable complex Hilbert space $\mathcal H$ and let $\mu$ be the scalar-valued spectral measure for the minimal normal extension $N$ of $S.$ Let $R^\infty (\sigma(S),\mu)$ be the weak-star closure in $L^\infty (\mu)$ of rational functions with poles off $\sigma(S),$ the spectrum of $S.$ The multiplier algebra $M(S)$ consists of functions $f\in L^\infty(\mu)$ such that $f(N)\mathcal H \subset \mathcal H.$ The multiplication operator $M_{S,f}$ of $f\in M(S)$ is defined $M_{S,f} = f(N) |_{\mathcal H}.$ We show that for $f\in R^\infty (\sigma(S),\mu),$ (1) $M_{S,f}$ is invertible iff $f$ is invertible in $M(S)$ and (2) $M_{S,f}$ is Fredholm iff there exists $f_0\in R^\infty (\sigma(S),\mu)$ and a polynomial $p$ such that $f=pf_0,$ $f_0$ is invertible in $M(S),$ and $p$ has only zeros in $\sigma (S) \setminus \sigma_e (S),$ where $\sigma_e (S)$ denotes the essential spectrum of $S.$ Consequently, we characterize $\sigma(M_{S,f})$ and $\sigma_e(M_{S,f})$ in terms of some cluster subsets of $f.$ Moreover, we show that if $S$ is an irreducible subnormal operator and $f \in R^\infty (\sigma(S),\mu),$ then $M_{S,f}$ is invertible iff $f$ is invertible in $R^\infty (\sigma(S),\mu).$ The results answer the second open question raised by J. Dudziak in 1984

Invertibility in Weak-Star Closed Algebras of Analytic Functions

For $K\subset \mathbb C$ a compact subset and $\mu$ a positive finite Bore1 measure supported on $K,$ let $R^\infty (K,\mu)$ be the weak-star closure in $L^\infty (\mu)$ of rational functions with poles off $K.$ We show that if $R^\infty (K,\mu)$ has no non-trivial $L^\infty$ summands and $f\in R^\infty (K,\mu),$ then $f$ is invertible in $R^\infty (K,\mu)$ if and only if Chaumat's map for $K$ and $\mu$ applied to $f$ is bounded away from zero on the envelope with respect to $K$ and $\mu.$ The result proves the conjecture $\diamond$ posed by J. Dudziak in 1984.Comment: arXiv admin note: text overlap with arXiv:2212.1081