Karabulut and Sibert (\textit{J. Math. Phys}. \textbf{38} (9), 4815 (1997)) have constructed an orthogonal set of functions from linear combinations of equally spaced Gaussians. In this paper we show that they are actually eigenfunctions of a q-oscillator in coordinate representation. We also reinterpret the coordinate representation example of q-oscillator given by Macfarlane as the functions orthogonal with respect to an unusual inner product definition. It is shown that the eigenfunctions in both q-oscillator examples are infinitely degenerate.
We study eigenfunctions of Schrodinger operators -y"+Py on the real line with zero boundary conditions, whose potentials P are real even polynomials with positive leading coefficients. For quartic potentials we prove that all zeros of all eigenfunctions belong to the union of the real and imaginary axes. Similar result holds for sextic potentials and their eigenfunctions with finitely many complex zeros. As a byproduct we obtain a complete classification of such eigenfunctions of sextic potentials.
We obtain sufficiently accurate eigenvalues and eigenfunctions for the anharmonic oscillator with potential $V(x,y)=x^{2}y^{2}$ by means of three different methods. Our results strongly suggest that the spectrum of this oscillator is discrete in agreement with early rigorous mathematical proofs and against a recent statement that cast doubts about it.
We prove that, given any finite link L in R^3, there is a high energy complex-valued eigenfunction of the harmonic oscillator such that its nodal set contains a union of connected components diffeomorphic to L. This solves a problem of Berry on the existence of knotted zeros in bound states of a quantum system.
We consider Gaussian random eigenfunctions (Hermite functions) of fixed energy level of the isotropic semi-classical Harmonic Oscillator on ${\bf R}^n$. We calculate the expected density of zeros of a random eigenfunction in the semi-classical limit $h \to 0.$ In the allowed region the density is of order $h^{-1},$ while in the forbidden region the density is of order $h^{-\frac{1}{2}}$. The computer graphics due to E.J. Heller illustrate this difference in "frequency" between the allowed and forbidden nodal sets.
In a recent paper [Phys. Lett. A 383 (2019) 2836; arXiv:1906.05121 [quant-ph]], Bagarello, Gargano, and Roccati have claimed that no square-integrable vacuum exists in quantizing the Bateman oscillator model. In this paper, we rebut their claim by actually deriving the square-integrable vacuum eigenfunction using a common procedure. We see that no problems occur in quantizing the Bateman oscillator model.
We consider the Schrödinger operator on the real line with an even quartic potential. Our main result is a product formula of the type $ψ_k(x)ψ_k(y) = \int_{\mathbb{R}} ψ_k(z)\mathcal{K}(x,y,z)dz$ for its eigenfunctions $ψ_k$. The kernel function $\mathcal{K}$ is given explicitly in terms of the Airy function $\mathrm{Ai}(x)$, and is positive for appropriate parameter values. As an application, we obtain a particular asymptotic expansion of the eigenfunctions $ψ_k$.
The aim of this paper is to provide an explicit expressions for the generalized q-deformed harmonic oscillator coherent states obtained in terms of a weak and strong behavior expansions. We first use the weak (s --> 0) deformed version of q-boson annihilation operator to solve Barut-Girardello's eigenvalues coherent states equation for the generalized q-deformed harmonic oscillator. In strong behavior limit (s --> 1) the previous result is resumed using the variational perturbation theory. We also describe the construction of their resolution of unity.
We consider an orthonormal basis of eigenfunctions of the Dirichlet Laplacian for a rational polygon. The modulus squared of the eigenfunctions defines a sequence of probability measures. We prove that this sequence contains a density-one subsequence that converges to Lebesgue measure.
We study a random Schroedinger operator, the Laplacian with N independently uniformly distributed random delta potentials on flat tori T^d_L = R^d/LZ^d, d = 2, 3, where L > 0 is large. We determine a condition in terms of the size of the torus L, the density of the potentials ρ= N/L^d and the energy of the eigenfunction E such any such eigenfunctions will with nonzero probability be uniformly distributed on the entire torus. We remark that the equidistribution we prove here is still consistent with a localized regime, where the localization length is much larger than the size of the torus. In …
We solve the eigenvalue problem of the $D_N$ type of Calogero model by mapping it to a set of decoupled quantum harmonic oscillators through a similarity transformation. In particular, we construct the eigenfunctions of this Calogero model from those of bosonic harmonic oscillators having either all even parity or all odd parity. It turns out that the eigenfunctions of this model are orthogonal with respect to a nontrivial inner product, which can be derived from the quasi-Hermiticity property of the corresponding conserved quantities.
In our previous paper I (del Valle--Turbiner, Int. J. Mod. Phys. A34, 1950143, 2019) it was developed the formalism to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As the result it was introduced - the Approximant - a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variationa…
The eigenvalues of the Liouvillians of Markovian master equation for a harmonic oscillator have a generic form. The Liouvillians considered are quadratic in the position coordinates or creation and annihilation operators, as well as having positive renormalized frequencies. We prove this by showing that a generic Liouvillian of this form can be similarly related to the Liouvillian of the Kossakowski--Lindblad equation, whose eigenvalues are already known. The left and right eigenfunctions of the generic Liouvillian also form a complete and biorthogonal set. Examples of similarly related right …
We study the scaling asymptotics of the eigenspace projection kernels $Π_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 Δ+ |x|^2$ of eigenvalue $E = \hbar(N + \frac{d}{2})$ in the semi-classical limit $\hbar \to 0$. The principal result is an explicit formula for the scaling asymptotics of $Π_{\hbar, E}(x,y)$ for $x,y$ in a $\hbar^{2/3}$ neighborhood of the caustic $\mathcal C_E$ as $\hbar \to 0.$ The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as $\hbar \to 0$. In previous work we proved that the den…
An explicit solution of the spectral problem of the non-local Schrödinger operator obtained as the sum of the square root of the Laplacian and a quartic potential in one dimension is presented. The eigenvalues are obtained as zeroes of special functions related to the fourth order Airy function, and closed formulae for the Fourier transform of the eigenfunctions are derived. These representations allow to derive further spectral propertiessuch as estimates of spectral gaps, heat trace and the asymptotic distribution of eigenvalues, as well as a detailed analysis of the eigenfunctions. A subtle…
We discuss new results on the geometry of eigenfunctions in disor- dered systems. More precisely, we study tori $R^d/LZ^d$, $d=2,3$, with uniformly distributed Dirac masses. Whereas at the bottom of the spectrum eigenfunctions are known to be localized, we show that for sufficiently large eigenvalue there exist uniformly distributed eigenfunctions with positive probability. We also study the limit $L\to\infty$ with a positive density of random Dirac masses, and deduce a lower polynomial bound for the localization length in terms of the eigenvalue for Poisson distributed Dirac masses on $R^d$ .…
We study many-body properties of quantum harmonic oscillator lattices with disorder. A sufficient condition for dynamical localization, expressed as a zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the eigenfunction correlators for an effective one-particle Hamiltonian. We show how state-of-the-art techniques for proving Anderson localization can be used to prove that these properties hold in a number of standard models. We also derive bounds on the static and dynamic correlation functions at both zero and positive temperature in terms of one-particle eigenfunction c…
Operator-theoretic approach to the partial integration of randomly coupled phase oscillators
Université Laval（拉瓦尔大学）；Centre interdisciplinaire en modélisation mathématique de l’Université Laval（拉瓦尔大学跨学科数学建模中心）；Centre de recherche CERVO（CERVO研究中心）
For the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$ the Perturbation Theory (PT) in powers of coupling constant $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) is developed constructively in $r$-space and in $(gr)$ space, respectively. The Riccati-Bloch (RB) equation and Generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate $r$-space and in $(gr)$-space, respectively, exploring the logarithmic derivative of wavefunction $y$. It is shown that PT in powers …
The convergence of the Rayleigh-Ritz method with nonlinear parameters optimized through minimization of the trace of the truncated matrix is demonstrated by a comparison with analytically known eigenstates of various quasi-solvable systems. We show that the basis of the harmonic oscillator eigenfunctions with optimized frequency ? enables determination of boundstate energies of one-dimensional oscillators to an arbitrary accuracy, even in the case of highly anharmonic multi-well potentials. The same is true in the spherically symmetric case of V (r) = ω2r2 2 + λrk, if k > 0. For spiked oscilla…
For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square}, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs respectively by Hans Lewy in 1977, and the authors in 2014 (see also Gauthier-Shalom--Przybytkowski, 2006). In this paper, we obtain similar results for the two dimensional isotropic quantum harmonic oscillator. In the opposite direction, we construct an infinite sequence of regu…
Considering symmetric strictly convex potentials, a local relationship is inferred from the virial theorem, based on which a real log-concave function can be constructed. Using this as a weight function and in such a way that the virial theorem can still be verified, parameter-free ansätze for the eigenfunctions of the associated Schrödinger equation are built. To illustrate the process, the technique is successfully tested against the harmonic oscillator, in which it leads to the exact eigenfunctions, and against the quartic anharmonic oscillator, which is considered the paradigmatic testing …
We revisit the radial oscillator from the free oscillator realization point of view. By using a free oscillator, namely the creation/annihilation operators of the harmonic oscillator, we construct an operator that maps the eigenfunctions of the harmonic oscillator to those of the radial oscillator. As a polynomial part of this relation, we obtain an operator that maps the Hermite polynomials to the Laguerre polynomials.
We report the identification and construction of raising and lowering operators for the complete eigenfunctions of isotropic harmonic oscillators confined by dihedral angles, in circular cylindrical and spherical coordinates; as well as for the hydrogen atom in the same situation of confinement, in spherical, parabolic and prolate spheroidal coordinates. The actions of such operators on any eigenfunction are examined in the respective coordinates, illustrating the possibility of generating the complete bases of eigenfunctions in the respective coordinates for both physical systems. The relatio…
Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of …
We introduce various optimization schemes for highly accurate calculation of the eigenvalues and the eigenfunctions of the one-dimensional anharmonic oscillators. We present several methods of analytically fixing the nonlinear variational parameter specified by the domain of the trigonometric basis functions. We show that the optimized parameter enables us to determine the energy spectrum to an arbitrary accuracy. Also, using the harmonic oscillator basis functions, we indicate that the resulting optimal frequency agrees with the one obtained by the principle of the minimal sensitivity.
In a previous paper a multi-species version of the q-Boson stochastic particle system is introduced and the eigenfunctions of its backward generator are constructed by using a representation of the Hecke algebra. In this article we prove a formula which expresses the eigenfunctions by means of the q-deformed bosonic operators, which are constructed from the L-operator of higher rank found in the recent work by Garbali, de Gier and Wheeler. The L-operator is obtained from the universal R-matrix of the quantum affine algebra of type A_{r}^{(1)} by the use of the q-oscillator representation. Thus…
In this note we consider a one-dimensional quantum mechanical particle constrained by a parabolic well perturbed by a Gaussian potential. As the related Birman-Schwinger operator is trace class, the Fredholm determinant can be exploited in order to compute the modified eigenenergies which differ from those of the harmonic oscillator due to the presence of the Gaussian perturbation. By taking advantage of Wang's results on scalar products of four eigenfunctions of the harmonic oscillator, it is possible to evaluate quite accurately the two lowest-lying eigenvalues as functions of the coupling c…
The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry-Keating hamiltonian H=(xp+px)/2 projected onto the subspace of oscillator eigenfunctions of lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert-Polya conjecture. The p-adic case is also discussed.
Comment on "Relativistic quantum oscillator model under the effects of the violation of Lorentz symmetry by an arbitrary fixed vector field'' by Faizuddin Ahmed
Facultad de Ciencias Naturales y Matemáticas, Universidad Nacional del Callao（卡亚俄国立大学自然科学与数学学院）；Universidade Federal do Maranhão（马拉尼昂联邦大学）；Universidad Tecnológica del Perú (UTP)（秘鲁理工大学）；Departamento de Ciencias, Universidad Privada del Norte (UPN)（北方私立大学科学系）
A system of functions (signals) on the finite line, called the oscillator system, is described and studied. Applications of this system for discrete radar and digital communication theory are explained. Keywords: Weil representation, commutative subgroups, eigenfunctions, random behavior, deterministic construction
