A full set of Casimir operators for the Lie superalgebra $gl(m/\infty)$ is constructed and shown to be well defined in the category $O_{FS}$ generated by the highest weight irreducible representations with only a finite number of non-zero weight components. The eigenvalues of these Casimir operators are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from $gl(m/\infty)$ are also determined.
The complete structure of the Casimir ${\cal{W}\cal{A}}_{\it{N}}$ algebras are shown to exist in such a way that the Casimir ${\cal{W}\cal{A}}_{\it{N}}$ algebra is a kind of truncated type of $\cal{W}_{\infty}$ algebra both in the primary and in the quadratic basis, first using the associativity conditions in the basis of primary fields and second using the Miura basis coming from the free field realization as a different basis. Finally one can say that the Casimir ${\cal{W}\cal{A}}_{\it{N}}$ algebra is a kind of truncated type of $\cal{W}_{\infty}$algebra,so it is clear from any construction …
Motivated by the universal knot polynomials in the gauge Chern-Simons theory, we show that the values of the second Casimir operator on an arbitrary power of Cartan product of $X_2$ and adjoint representations of simple Lie algebras can be represented in a universal form. We show that it complies with $N\longrightarrow -N$ duality of the same operator for $SO(2n)$ and $Sp(2n)$ algebras (the part of $N\leftrightarrow-N$ duality of gauge $SO(2n)$ and $Sp(2n)$ theories). We discuss the phenomena of non-zero universal values of Casimir operator on zero representations.
Using the group theoretic method of spectrum generating algebras a class of differential equations is obtained whose eigenvalues are calculated without explicitly solving the equations. Solutions can be easily obtained by group theoretic methods for a certain type of potentials.
The nonstandard q-deformed algebras U'_q(so_n) are known to possess q-analogues of Gel'fand-Tsetlin type representations. For these q-algebras, all the Casimir elements (corresponding to basis set of Casimir elements of so(n)) are found, and their eigenvalues within irreducible representations are given explicitly.
It is given a way of computing Casimir eigenvalues for Weyl orbits as well as for irreducible representations of Lie algebras. A kappa(s) number of polinomials which depend on rank N are obtained explicitly for A_N Casimir operators of order s where kappa(s) is the number of partitions of s into positive integers except 1. It is also emphasized that these eigenvalue polinomials prove useful in obtaining formulas to calculate weight multiplicities and in explicit calculations of the whole cohomology ring of Classical and also Exceptional Lie algebras.
A full set of (higher order) Casimir invariants for the Lie algebra $gl(\infty )$ is constructed and shown to be well defined in the category $O_{FS}$ generated by the highest weight (unitarizable) irreducible representations with only a finite number of non-zero weight components. Moreover the eigenvalues of these Casimir invariants are determined explicitly in terms of the highest weight. Characteristic identities satisfied by certain (infinite) matrices with entries from $gl(\infty )$ are also determined and generalize those previously obtained for $gl(n)$ by Bracken and Green.$^{1,2}$
This paper deals with the striking fact that there is an essentially canonical path from the $i$-th Lie algebra cohomology cocycle, $i=1,2,... l$, of a simple compact Lie algebra $\g$ of rank $l$ to the definition of its primitive Casimir operators $C^{(i)}$ of order $m_i$. Thus one obtains a complete set of Racah-Casimir operators $C^{(i)}$ for each $\g$ and nothing else. The paper then goes on to develop a general formula for the eigenvalue $c^{(i)}$ of each $C^{(i)}$ valid for any representation of $\g$, and thereby to relate $c^{(i)}$ to a suitably defined generalised Dynkin index. The for…
The uniformity, for the family of exceptional Lie algebras g, of the decompositions of the powers of their adjoint representations is well-known now for powers up to the fourth. The paper describes an extension of this uniformity for the totally antisymmetrised n-th powers up to n=9, identifying (see Tables 3 and 6) families of representations with integer eigenvalues 5,...,9 for the quadratic Casimir operator, in each case providing a formula (see eq. (11) to (15)) for the dimensions of the representations in the family as a function of D=dim g. This generalises previous results for powers j …
We give a general method to construct a complete set of linearly independent Casimir operators of a Lie algebra with rank N. For a Casimir operator of degree p, this will be provided by an explicit calculation of its symmetric coefficients $ g^{A_1,A_2,.. A_p}$. It is seen that these coefficients can be descibed by some rational polinomials of rank N. These polinomials are also multilinear in Cartan sub-algebra indices taking values from the set $I_0 = {1,2,.. N}$. The crucial point here is that for each degree one needs, in general, more than one polinomials. This in fact is related with an o…
In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another important invariant besides the Hamiltonian for each invariant subsystem is found, and is utilized to prove an ``unstable disk theorem'' through a simple Energy-Casimir argument. The eigenvalues of the linear Hamiltonian system are of four types: real pairs ($c,-c$), purely imaginary pairs ($id,-id$), quadruples ($\pm c\pm id$), and zero eigenvalues. The eigenva…
The Hilbert space of the unitary irreducible representations of a Lie group that is a quantum dynamical group are identified with the quantum state space. Hermitian representation of the algebra are observables. The eigenvalue equations for the representation of the set of Casimir invariant operators define the field equations of the system. A general class of dynamical groups are semidirect products K *s N for which the representations are given by Mackey's theory. The homogeneous group K must be a subgroup of the automorphisms of the normal group N. The archetype dynamical group is the Poinc…
There has been proposed a new method of the constructing of the basic functions for spaces of tensor representations of the Lie groups with the help of the generalized Casimir operator. In the definition of the operator there were used the Lie derivatives instead of the corresponding infenitisemal operators. When introducing the generalized Casimir operator we use the metric for which a group being considered will be isometry that follows from the invariance condition for the generalized Casimir operator. This allows us to formulate the eigenvalue and eigenfunction problems correctly. The inva…
Let $H=-D^2+V$ be a Schrödinger operator on $ L^2(\mathbb{R})$, or on $ L^2(0,\infty)$. Suppose the potential satisfies $\limsup_{x\to \infty}|xV(x)|=a<\infty$. We prove that $H$ admits no eigenvalue larger than $ \frac{4a^2}{π^2}$. For any positive $a$ and $λ$ with $0<λ< \frac{4a^2}{π^2}$, we construct potentials $V$ such that $\limsup_{x\to \infty}|xV(x)|=a $ and the associated Schrodinger operator $H=-D^2+V$ has eigenvalue $λ$.
Let $H_0$ be a periodic operator on $\R^+$(or periodic Jacobi operator on $\N$). It is known that the absolutely continuous spectrum of $H_0$ is consisted of spectral bands $\cup[α_l,β_l]$. Under the assumption that $\limsup_{x\to \infty} x|V(x)|<\infty$ ($\limsup_{n\to \infty} n|V(n)|<\infty$), the asymptotical behaviour of embedded eigenvalues approaching to the spectral boundary is established.
It is shown, for any irreducible representation of $E_8$ Lie algebra, that eigenvalues of Casimir operators can be calculated in the form of invariant polinomials which are decomposed in terms of $A_8$ basis functions. The general method is applied for degrees 8,12 and 14 for which 2,8 and 19 invariant polinomials are obtained respectively. For each particular degree, these invariant polinomials can be taken to be $E_8$ basis functions in the sense that any Casimir operator of $E_8$ has always eigenvalues which can be expressed as linear superpositions of them. This can be investigated by show…
In this paper the quantum vacuum energies induced by massive fluctuations of one real scalar field on a configuration of two partially transparent plates are analysed. The physical properties of the infinitely thin plates are characterized by two Dirac-$δ$ potentials. We find that an attractive/repulsive Casimir force arises between the plates when the weights of the $δ$'s have equal/different sign. If some of the plates absorbs fluctuations below some threshold of energy (the corresponding weight is negative) there is the need to set a minimum mass to the scalar field fluctuations to preserve…
A common tool in Casimir physics (and many other areas) is the asymptotic (high-frequency) expansion of eigenvalue densities, employed as either input or output of calculations of the asymptotic behavior of various Green functions. Here we clarify some fine points and potentially confusing aspects of the subject. In particular, we show how recent observations of Kolomeisky et al. [Phys. Rev. A 87 (2013) 042519] fit into the established framework of the distributional asymptotics of spectral functions.
We discuss the eigenvalues $E_j$ of Schrödinger operators $-Δ+V$ in $L^2(\mathbb R^d)$ with complex potentials $V\in L^p$, $p<\infty$. We show that (A) $\mathrm{Re} E_j\to\infty$ implies $\mathrm{Im} E_j\to 0$, and (B) $\mathrm{Re} E_j\to E\in [0,\infty)$ implies $(\mathrm{Im} E_j)\in\ell^q$ for some $q$ depending on $p$. We prove quantitative versions of (A) and (B) in terms of the $L^p$-norm of $V$.
An asymptotic expression of the orthonormal polynomials $\mathcal{P}_{N}(z)$ as $N\rightarrow\infty$, associated with the singularly perturbed Laguerre weight $w_α(x;t)=x^α{\rm e}^{-x-\frac{t}{x}},~x\in[0,\infty),~α>-1,~t\geq0$ is derived. Based on this, we establish the asymptotic behavior of the smallest eigenvalue, $λ_{N}$, of the Hankel matrix generated by the weight $w_α(x;t)$.
We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2n\times 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,2k}=\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,0}= -\frac{1}{\sqrt{2π}}ζ\left(\frac{3}{2}\right),$$ where $ζ$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n)_{n\geq 1}$, $$\lim_{n\rightarrow \infty}\frac {1}{\sqrt{2n}} \log p_{2n,2k_n}=-\frac{1}{\sqrt{2π}}ζ\left(\frac{3}{2}\right),$$ provided $\lim_{n\right…
The class of differential-equation eigenvalue problems $-y''(x)+x^{2N+2}y(x)=x^N Ey(x)$ ($N=-1,0,1,2,3,...$) on the interval $-\infty<x<\infty$ can be solved in closed form for all the eigenvalues $E$ and the corresponding eigenfunctions $y(x)$. The eigenvalues are all integers and the eigenfunctions are all confluent hypergeometric functions. The eigenfunctions can be rewritten as products of polynomials and functions that decay exponentially as $x\to\pm \infty$. For odd $N$ the polynomials that are obtained in this way are new and interesting classes of orthogonal polynomials. For example, w…
We investigate the spectrum of Schrödinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to $-\infty$, a narrow cluster of finitely many eigenvalues tends to $-\infty$, while the eigenvalues above the cluster remain bounded from below. Certain "rogue" eigenvalues break away from this cluster and tend even faster toward $-\infty$. The spectrum can be visualized as the intersection points of two objects in the plane--a spiral curve depending on the Schrödinger potential, and a set o…
We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using $2(2^{N-1}-1)$ bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labelled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy…
For integers $m\geq 3$, we study the non-self-adjoint eigenvalue problems $-u^{\prime\prime}(x)+(x^m+P(x))u(x)=E u(x)$, $0\leq x<+\infty$, with the boundary conditions $u(+\infty)=0$ and $αu(0)+βu^{\prime}(0)=0$ for some $α, β\in\C$ with $|α|+|β|\not=0$, where $P(x)=a_1 x^{m-1}+a_2 x^{m-2}+...+a_{m-1} x$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $E_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials from asymptotic expansions of the eigenvalues.
We present an extension of the trace of a vertex operator and explain a representation-theoretic interpretation of the trace. Specifically, we consider a twist of the vertex operator with infinitely many Casimir operators and compute its trace as a character formula. To do this, we define the Fock space of infinite level $\mathfrak{F}^{\infty}$. Then, we prove a duality between $\mathfrak{gl}_{\infty}$ and $\mathfrak{a}_{\infty}=\widehat{\mathfrak{gl}}_{\infty}$ of Howe type, which provides a decomposition of $\mathfrak{F}^{\infty}$ into irreducible representations with joint highest weight ve…
We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the $n\times n$ Laguerre Unitary Ensemble. The probability that this ratio is greater than $r>1$ is expressed in terms of an $n \times n$ Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as $n\to\infty$ is found as an integral over $(0,\infty)$ containing two functions $q_{1}(x)$ and $q_{2}(x)$. These functions satisfy a system of two coupled Painlevé V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. We comp…
We study the first eigenvalue of the $p-$Laplacian (with $1<p<\infty$) on a quantum graph with Dirichlet or Kirchoff boundary conditions on the nodes. We find lower and upper bounds for this eigenvalue when we prescribe the total sum of the lengths of the edges and the number of Dirichlet nodes of the graph. Also we find a formula for the shape derivative of the first eigenvalue (assuming that it is simple) when we perturb the graph by changing the length of an edge. Finally, we study in detail the limit cases $p\to \infty$ and $p\to 1$.
It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix $V$ is concave (convex) with respect to $V$. Using the theory of the spectral shift function we generalize this property to self-adjoint operators on a separable Hilbert space with an arbitrary spectrum. More precisely, we prove that the spectral shift function integrated with respect to the spectral parameter from $-\infty$ to $λ$ (from $λ$ to $+\infty$) is concave (convex) with respect to trace class perturbations. The case of relative trace class perturbations is also considered.
We study Schrödinger operators $H=-Δ+V$ in $L^2(Ω)$ where $Ω$ is $\mathbb R^d$ or the half-space $\mathbb R_+^d$, subject to (real) Robin boundary conditions in the latter case. For $p>d$ we construct a non-real potential $V\in L^p(Ω)\cap L^{\infty}(Ω)$ that decays at infinity so that $H$ has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum $σ_{\rm ess}(H)=[0,\infty)$. This demonstrates that the Lieb-Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case.
We present a method to derive asymptotics of eigenvalues for trace-class integral operators $K:L^2(J;dλ)\circlearrowleft$, acting on a single interval $J\subset\mathbb{R}$, which belong to the ring of integrable operators \cite{IIKS}. Our emphasis lies on the behavior of the spectrum $\{λ_i(J)\}_{i=0}^{\infty}$ of $K$ as $|J|\rightarrow\infty$ and $i$ is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant $\det(I-γK)|_{L^2(J)}$ as $|J|\rightarrow\infty$ and $γ\uparrow 1$ in a Stokes type scaling regime. Concrete asymptotic formulæ\, are obtained f…
