Related papers: Exceptional points of the eigenvalues of parameter…
A Hamiltonian operator $\hat H$ is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical…
Exceptional points are the branch-point singularities of non-Hermitian Hamiltonians, and have rich consequences in open-system dynamics. While the exceptional points and their critical phenomena are widely studied in the non-Hermitian…
This paper deals with eigenvalues and eigenvectors of bicomplex linear operators defined on bicomplex space. We investigate the properties of these operators in the context of eigenvalues and eigenvectors, along with some relevant theorems.…
Starting from the expression for the superdeterminant of (xI-M), where M is an arbitrary supermatrix, we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic…
We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear…
We deduce the eigenvalues and the eigenvectors of a parameter-dependent Hamiltonian $H_\theta$ which is closely related to the Swanson Hamiltonian, and we construct bi-coherent states for it. After that, we show how and in which sense the…
We show how to derive fixed-point Hamiltonians in quantum mechanics from a proposed renormalization group invariance approach that relies in a subtraction procedure at a given energy scale. The scheme is valid for arbitrary interactions…
A polynomial is expansive if all of its roots lie outside the unit circle. We define some special determinants involving the coefficients of a real polynomial and formulate necessary and sufficient conditions for expansivity using these…
We reveal a novel topological property of the exceptional points in a two-level parity-time symmetric system and then propose a scheme to detect the topological exceptional points in the system, which is embedded in a larger Hilbert space…
We prove quantitative bounds on the eigenvalues of non-selfadjoint bounded and unbounded operators. We use the perturbation determinant to reduce the problem to one of studying the zeroes of a holomorphic function.
The question of defining unique, generally applicable constrained second, and higher-order, derivatives is investigated. It is shown that second-order constrained derivatives obtained via two successive constrained differentiations provide…
The Macdonald finite-difference Hamiltonian is lifted to a super-generalization. In addition to canonical bosonic time variables $p_k$ new Grassmann time variables $\theta_k$ are introduced, and the Hamiltonian is represented as a…
A well-known characterization of Jordan vectors of a matrix polynomial $L(z)$ is generalized to a characterization of Jordan vectors of the operator-valued function $Q(z)$ at an eigenvalue $\alpha \in \mathbb{C}$. The results are then…
We consider self-similar measures on $\mathbb R.$ The Hutchinson operator $H$ acts on measures and is the dual of the transfer operator $T$ which acts on continuous functions. We determine polynomial eigenfunctions of $T .$ As a…
We study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological…
In this paper we study the index of reducibility of powers of a standard parameter ideal. An explicit formula is proved for the extremely case. We apply the main result to compute Hilbert polynomials of socle ideals of standard parameter…
The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…
We provide the mathematical foundation for the $X_m$-Jacobi spectral theory. Namely, we present a self-adjoint operator associated to the differential expression with the exceptional $X_m$-Jacobi orthogonal polynomials as eigenfunctions.…
The Lindblad equation for a two-level system under an electric field is analyzed by mapping to a linear equation with a non-Hermitian matrix. Exceptional points of the matrix are found to be extensive; the second-order ones are located on…
We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we…