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The connection between spherical harmonics and symmetric tensors is explored. For each spherical harmonic, a corresponding traceless symmetric tensor is constructed. These tensors are then extended to include nonzero traces, providing an…
This study is devoted to the asymptotic spectral analysis of multiscale Schr\"odinger operators with oscillating and decaying electric potentials. Different regimes, related to scaling considerations, are distinguished. By means of a normal…
We consider the $q$-deformed Schr\"odinger equation of the harmonic oscillator on the $N$-dimensional quantum Euclidian space. The creation and annihilation operator are found, which systematically produce all energy levels and…
Recently, an application of the numerical bootstrap method to quantum mechanics was proposed, and it successfully reproduces the eigenstates of various systems. However, it is unclear why this method works. In order to understand this…
We study the semirelativistic Hamiltonian operator composed of the relativistic kinetic energy and a static harmonic-oscillator potential in three spatial dimensions and construct, for bound states with vanishing orbital angular momentum,…
In this paper we give a general solution to the problem of the damped harmonic oscillator under the influence of an arbitrary time-dependent external force. We employ simple methods accessible for beginners and useful for undergraduate…
The Schr\"odinger operator on a metric tree is a family of ordinary differential operators on its edges complemented by certain matching conditions at the vertices. The regular trees are highly symmetric. This allows one to construct an…
Let L be a Schr\"odinger operator of the form L=-\Delta+V, where the nonnegative potential V satisfies a reverse H\"older inequality. Using the method of L-harmonic extensions we study regularity estimates at the scale of adapted H\"older…
Type III multi-step rationally-extended harmonic oscillator and radial harmonic oscillator potentials, characterized by a set of $k$ integers $m_1$, $m_2$, \ldots, $m_k$, such that $m_1 < m_2 < \cdots < m_k$ with $m_i$ even (resp.\ odd) for…
In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A_n(z) and B_n(z), appearing in the ladder operators satisfy the two…
The annihilation-creation operators of the harmonic oscillator, the basic and most important tools in quantum physics, are generalised to most solvable quantum mechanical systems of single degree of freedom including the so-called…
In this paper we generalize the spin-raising and lowering operators of spin-weighted spherical harmonics to linear-in-$\gamma$ spin-weighted spheroidal harmonics where $\gamma$ is an additional parameter present in the second order ordinary…
In this article, we give the explicit solutions to the Laplace equations associated to the Dirac operator, Euler operator and the harmonic oscillator in R.
We construct the coherent states of general order, $m$ for the ladder operators, $c(m)$ and $c^\dagger(m)$, which act on rational deformations of the harmonic oscillator. The position wavefunctions of the eigenvectors involve type III…
This paper is devoted to the spectral theory of the Schr\"{o}dinger operator on the simplest fractal: Dyson's hierarchical lattice. An explicit description of the spectrum, eigenfunctions, resolvent and parabolic kernel are provided for the…
A quantum realization of the Relativistic Harmonic Oscillator is realized in terms of the spatial variable $x$ and ${\d\over \d x}$ (the minimal canonical representation). The eigenstates of the Hamiltonian operator are found (at lower…
We study arbitrary order symmetry operators for the linear Schr\"odinger equations with arbitrary number of spatial variables. We deduce determining equations for coefficient functions of such operators and consider in detail some cases…
Schur multipliers are basic linear maps on matrix algebras. Their close albeit still intriguing connection with Fourier multipliers establishes a powerful bridge between harmonic analysis and operator algebras. In this paper, we survey…
Superoscillations are a phenomenon in physics, where linear combinations of low-frequency plane waves interfere almost destructively in such a way that the resulting wave has a higher frequency than any of the individual waves. The…
Using the modified factorization method employed by Mielnik for the harmonic oscillator, we show that isospectral structures associated with a second order operator $H$, can always be constructed whenever $H$ could be factored, or exist…