On Infinite Discrete Spectrum of Convolution Operators with Potentials

The goal of this note is to study the spectrum of a self-adjoint convolution operator in $L^2(\mathbb R^d)$ with an integrable kernel that is perturbed by an essentially bounded real-valued potential tending to zero at infinity. We show that the essential spectrum of such operator is the union of the spectrum of the convolution operator and of the essential range of the potential. Then we provide several sufficient conditions for the existence of a countable sequence of discrete eigenvalues. For operators having non-connected essential spectrum we give sufficient conditions for the existence of discrete eigenvalues in the corresponding spectral gaps.

We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary of the domain. We show that there exists a critical total magnetic flux depending on the aperture of the conical surface for which the system undergoes an abrupt spectral transition from infinitely many eigenvalues below the essential spectrum to an empty discrete spectrum. For the critical flux, we establish a Hardy-type inequality. In the regime with an infinite discrete spectrum, we obtain sharp spectral asymptotics with a refined estimate of the remainder and investigate the dependence of the eigenvalues on the aperture of the surface and the flux of the magnetic field.

We introduce a one-parameter-generalized oscillator algebra (that covers the case of the harmonic oscillator algebra) and discuss its finite- and infinite-dimensional representations according to the sign of the parameter κ. We define an (Hamiltonian) operator associated with and examine the degeneracies of its spectrum. For the finite (when κ < 0) and the infinite (when κ ⩾ 0) representations of , we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators. To overcome the difficulties related to the phase operator in the infinite-dimensional case and to avoid the degeneracy problem for the finite-dimensional case, we introduce a truncation procedure which generalizes the one used by Pegg and Barnett for the harmonic oscillator. This yields a truncated-generalized oscillator algebra , where s denotes the truncation order. We construct two types of temporally stable states for (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of ). Two applications are considered in this paper. The first concerns physical realizations of and in the context of one-dimensional quantum systems with finite (Morse system) or infinite (Pöschl–Teller system) discrete spectra. The second deals with mutually unbiased bases used in quantum information.

The multiphoton algebras for one-dimensional Hamiltonians with infinite discrete spectrum, and for their associated kth-order SUSY partners are studied. In both cases, such an algebra is generated by the multiphoton annihilation and creation operators, as well as by Hamiltonians which are functions of an appropriate number operator. The algebras obtained turn out to be polynomial deformations of the corresponding single-photon algebra previously studied in literature. The Barut-Girardello coherent states, which are eigenstates of the annihilation operator, are obtained and their uncertainty relations are explored by means of the associated quadratures.

We investigate the one-dimensional Schrodinger operator. The condition that the potential be self-similar under Darboux transformation leads to transparent potentials with infinitely many eigenvalues.

In this work, we consider a general Morse-type confining potential that was introduced recently by Alhaidari (J Theor Math Phys 205, 2020. arXiv:2005.09080 ). This potential is completely confining and hence has an infinite discrete spectrum. We compute the energy spectrum associated with this confining potential using two different approaches so as to ensure the correctness of our results. We use both asymptotic iteration method and the numerical diagonalization method based on the tridiagonal representation approach of our unperturbed Hamiltonian to compute the energy spectrum. We found that both approaches resulted in bound state energies that are in agreement with each other to a high degree of accuracy.

We construct infinitely many examples of pairs of isospectral but non-isometric [Formula: see text]-cusped hyperbolic [Formula: see text]-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada’s method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.

We construct static and time dependent exact soliton solutions for a theory of scalar fields taking values on a wide class of two dimensional target spaces, and defined on the four dimensional space–time S3×R. The construction is based on an ansatz built out of special coordinates on S3. The requirement for finite energy introduce boundary conditions that determine an infinite discrete spectrum of frequencies for the oscillating solutions. For the case where the target space is the sphere S2, we obtain static soliton solutions with nontrivial Hopf topological charges. In addition, such Hopfions can oscillate in time, preserving their topological Hopf charge, with any of the frequencies belonging to that infinite discrete spectrum.

General sets of coherent states are constructed for quantum systems admitting a nondegenerate infinite discrete energy spectrum. They are eigenstates of an annihilation operator and satisfy the usual properties of standard coherent states. The application of such a construction to the quantum optics Jaynes–Cummings model leads to a new understanding of the properties of this model.

We investigate the discrete spectrum of the Schr?dinger operator H for a system of three particles. We assume that the operators h?, ? = 1,?2,?3, which describe the three subsystems of two particles do not have any negative eigenvalues. Under the assumption that either two or three of the operators h? have so-called virtual levels at the start of the continuous spectrum, we establish the existence of an infinite discrete spectrum for the three-particle operator H. The functions which describe the interactions between pairs of particles can be rapidly decreasing (or even of compact support) with respect to x. Bibliography: 17 items.

По выбранной ортогональной последовательности полиномов $\{p_n(s)\}_{n=0}^{\infty}$, которая имеет дискретный спектр, строится энергетический спектр $E_k=f(s_k)$, где $\{s_k\}$ - точки конечного или бесконечного дискретного спектра полиномов. С помощью подхода к квантовой механике, который основан не на функциях потенциала, а на ортогональных полиномах, зависящих от энергии, построена локальная численная реализация потенциала, отвечающая выбранному энергетическому спектру. В качестве примера рассмотрены трехпараметрические непрерывные дуальные полиномы Хана. Приведены точные аналитические выражения для соответствующего энергетического спектра связанных состояний, сдвига фазы состояний рассеяния и волновых функций. Однако функция потенциала для заданного набора физических параметров получается только численно.

Analysis on the minimal representation of O( p, q) II. Branching laws

Sinusoidal signals and complex exponentials play a critical role in LTI system theory in that they are eigenfunctions of the LTI convolution operator. While processing frequency-modulated (FM) waveforms using LTI systems, restrictive assumptions must be placed on the system so that a quasi-eigenfunction approximation holds. Upon deviation from these assumptions, FM waveforms incur significant distortion. In this paper, a Sturm-Liouville (S-L) model for frequency modulation introduced by the author, is extended to a) study orthogonal modes of continuous and discrete frequency modulation and b) to develop system theoretical underpinnings for FM waveforms. These FM modes have the same special connection with respect to the FM S-L system operator, that complex exponentials have with LTI systems and the convolution operator. The finite S-L-FM spectrum or transform that measures the strength of the orthogonal FM modes present in a FM signal, analogous to the discrete Fourier spectrum for sinusoids, is introduced. Finally, similarities between the orthogonal S-L-FM modes and angular Mathieu functions are exposed, and a conjecture connecting the two is put forth.

ABSTRACTWe study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.

This paper focuses on the spectral properties of a bounded self-adjoint operator in L2(Rd) being the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. We study both the essential and the discrete spectra of this operator. It is shown that the essential spectrum of the sum is the union of the essential spectrum of the convolution operator and the image of the potential. We then provide a number of sufficient conditions for the existence of discrete spectrum and obtain lower and upper bounds for the number of discrete eigenvalues. Special attention is paid to the case of operators possessing countably many points of the discrete spectrum. We also compare the spectral properties of the operators considered in this work with those of classical Schrödinger operators.