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 and Casimir eigenvalues j, j<=4. Many intriguing, perhaps puzzling, features of the dimension formulas are discussed and the possibility that they may be valid for a wider class of not necessarily simple Lie algebras is considered.
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.
A formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second-order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional-type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen-van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar…
Exceptional points of a class of non-hermitian Hamilton operators $\hat H$ of the form $\hat H=\hat H_0+i\hat H_1$ are studied, where $\hat H_0$ and $\hat H_1$ are hermitian operators. Finite dimensional Hilbert spaces are considered. The linear operators $\hat H_0$ and $\hat H_1$ are given by spin matrices for spin $s=1/2,1,3/2,\dots$. Since the linear operators studied are nonnormal, properties of such operators are described.
The purpose of this paper is to study the exceptional eigenvalues of the asymmetric quantum Rabi models (AQRM), specifically, to determine the degeneracy of their eigenstates. Here, the Hamiltonian $H^ε_{\text{Rabi}}$ of the AQRM is defined by adding the fluctuation term $εσ_x$, with $σ_x$ being the Pauli matrix, to the Hamiltonian of the quantum Rabi model, breaking its $\mathbb{Z}_{2}$-symmetry. The spectrum of $H^ε_{\text{Rabi}}$ contains a set of exceptional eigenvalues, considered to be remains of the eigenvalues of the uncoupled bosonic mode, which are further classified in two types: Ju…
Canonically relativistic quantum mechanics: Casimir field equations of the quaplectic group
量子动力学群李群的酉不可约表示的希尔伯特空间与量子态空间相同。代数的厄米特表示是可观的。表示Casimir不变算子集的本征值方程定义了系统的场方程。一类一般的动力群是半直积K * s N，其表示由麦基理论给出。齐次群K必须是正规群N的自同构的子群。原型动力群是庞加莱群。由卡西米尔算符的表象定义的场方程定义了物理学的基本方程;克莱因-戈登方程、狄拉克方程、麦克斯韦方程等等。本文探讨了一个更一般的动力群候选者，也是一个半直积，但其中的“翻译”正常子群N现在是海森堡群。海森堡群的相关自同构是辛群。这与对正交度量的要求一起导致齐次群K的伪酉群。给出了这个群的物理意义和动机，称之为四元群，并确定了该代数的厄米特不可约表示。与庞加莱群一样，群的选择定义了与量子粒子态相一致的表示的希尔伯特空间。给出了Casimir算子表示的本征值方程--场方程，并对其进行了研究。该理论体现了玻恩互易原理和一个新的相对性原理。
We study a three-parameter family of PT-symmetric Hamiltonians, related via the ODE/IM correspondence to the Perk-Schultz models. We show that real eigenvalues merge and become complex at quadratic and cubic exceptional points, and explore the corresponding Jordon block structures by exploiting the quasi-exact solvability of a subset of the models. The mapping of the phase diagram is completed using a combination of numerical, analytical and perturbative approaches. Among other things this reveals some novel properties of the Bender-Dunne polynomials, and gives a new insight into a phase trans…
In this review paper we carry on our investigations on Schroedinger operators with inverse square potentials on the half-line. Depending on several parameters, such operators possess either a finite number of complex eigenvalues, or an infinite one, but also some spectral singularities embedded in the continuous spectrum (exceptional situations). The spectral and the scattering theory for these operators is recalled, and new results for the exceptional cases are provided. Some index theorems in scattering theory are also developed, and explanations why these results can not be extended to the …
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.
