This paper presents the spectral analysis of 1-dimensional Schroedinger operator on the half-line whose potential is a linear combination of the Coulomb term 1/r and the centrifugal term 1/r^2. The coupling constants are allowed to be complex, and all possible boundary conditions at 0 are considered. The resulting closed operators are organized in three holomorphic families. These operators are closely related to the Whittaker equation. Solutions of this equation are thoroughly studied in a large appendix to this paper. Various special cases of this equation are analyzed, namely the degenerate, the Laguerre and the doubly degenerate cases. A new solution to the Whittaker equation in the doubly degenerate case is also introduced.
This paper presents a thorough analysis of 1-dimensional Schroedinger operators whose potential is a linear combination of the Coulomb term 1/r and the centrifugal term 1/r^2. We allow both coupling constants to be complex. Using natural boundary conditions at 0, a two parameter holomorphic family of closed operators is introduced. We call them the Whittaker operators, since in the mathematical literature their eigenvalue equation is called the Whittaker equation. Spectral and scattering theory for Whittaker operators is studied. Whittaker operators appear in quantum mechanics as the radial pa…
An infinite family of closed-form solutions is exhibited for the Schroedinger equation for the potential $V(r) = -Z/\sqrt{|r|^{2} + a^{2}}$. Evidence is presented for an approximate dynamical symmetry for large values of the angular momentum $l$.
Consider the Schroedinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator's resolvent at a positive energy E are bounded by O(1/h) if and only if the associated Hamilton flow is non-trapping at energy E. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.
A recently proposed algorithm to obtain global solutions of the double confluent Heun equation is applied to solve the quantum mechanical problem of finding the energies and wave functions of a particle bound in a potential sum of a repulsive supersingular term, Ar(-4), plus an attractive Coulombian one, -Zr(-1). The existence of exact algebraic solutions for certain values of A is discussed.
The aim of the paper is to derive spectral estimates on the eigenvalue moments of the magnetic Schroedinger operators defined on the two-dimensional disk with a radially symmetric magnetic field and radially symmetric electric potential.
The Schroedinger equation on the half line is considered with a real-valued, integrable potential having a finite first moment. It is shown that the potential and the boundary conditions are uniquely determined by the data containing the discrete eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that boundary condition, and a subset of the discrete eigenvalues for a different boundary condition. This result extends the celebrated two-spectrum uniqueness theorem of Borg and Marchenko to the case where there is also a continuous spectrum
We consider radial tree extensions of one-dimensional quasi-periodic Schroedinger operators and establish the stability of their absolutely continuous spectra under weak but extensive perturbations by a random potential. The sufficiency criterion for that is the existence of Bloch-Floquet states for the one dimensional operator corresponding to the radial problem.
The method of potential envelopes is used to analyse the bound-state spectrum of the Schroedinger Hamiltonian H = -Delta -v/(r+b), where v and b are positive. We established simple formulas yielding upper and lower energy bounds for all the energy eigenvalues.
We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree-Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray Free Electron Laser (XFEL). We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding …
In this paper we consider magnetic Schroedinger operators on the two-dimensional unit disk with a radially symmetric magnetic field which explodes to infinity at the boundary. We prove a bound for the eigenvalue moments and a bound for the number of negative eigenvalues for such operators.
We give a short description of the proof of asymptotic-completeness for NLS-type equations, including time dependent potential terms, with radial data in three dimensions. We also show how the method applies for the two-body Quantum Scattering case.
We study the heat semigroup generated by two-dimensional Schroedinger operators with compactly supported magnetic field. We show that if the field is radial, then the large time behavior of the associated heat kernel is determined by its total flux. We also establish some on-diagonal heat kernel estimates and discuss their applications for solutions to the heat equation. An exact formula for the heat kernel, and for its large time asymptotic, is derived in the case of the Aharonov-Bohm magnetic field.
For non-relativistic Schroedinger equations the lowering of their degree by substitution Psi(r) \to F(r) =Psi'(r) /Psi(r) is known to facilitate our understanding and use of their (incomplete, so called quasi-exact) solvability. We show that and how the radial Dirac relativistic equation may quasi-exactly be solved in similar spirit.
We study the radial part of the Dunkl-Coulomb problem in two dimensions and show that this problem possesses the $su(1,1)$ symmetry. We introduce two different realizations for the $su(1,1)$ Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and the eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl-Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, been one of them proportional to the radial Hamiltonian. Finally, we use the …
Based on the Caputo fractional derivative the classical, non relativistic Hamiltonian is quantized leading to a fractional Schroedinger type wave equation. The free particle solutions are localized in space. Solutions for the infinite well potential and the radial symmetric ground state solution are presented. It is shown, that the behaviour of these functions may be reproduced with a ordinary Schroeodinger equation with an additional potential, which is of the form V ~ x for $α<1$, corresponding to the confinement potential, introduced phenomenologically to the standard models for non relativ…
We discuss the separability of the position-dependent mass Hamiltonian in cylindrical coordinates in the framework of a radial power-law position-dependent mass. We consider two particular radial mass settings; a harmonic oscillator type, and a Coulombic type. We subject the radial harmonic oscillator type mass to a radial harmonic oscillator potential and the radial Coulombic mass to a radial Coulombic potential. Azimuthal symmetry is assumed and spectral signatures of various z-dependent interaction potentials are reported.
We study the question of magnetic confinement of quantum particles on the unit disk $\ID$ in $\IR^2$, i.e. we wish to achieve confinement solely by means of the growth of the magnetic field $B(\vec x)$ near the boundary of the disk. In the spinless case we show that $B(\vec x)\ge \frac{\sqrt 3}{2}\cdot\frac{1}{(1-r)^2}-\frac{1}{\sqrt 3}\frac{1}{(1-r)^2\ln \frac{1}{1-r}}$, for $|\vec x|$ close to 1, insures the confinement provided we assume that the non-radially symmetric part of the magnetic field is not very singular near the boundary. Both constants $\frac{\sqrt 3}{2}$ and $-\frac{1}{\sqrt …
We use the Hellman-Feynman (HF) and Hypervirial (HV) theorems, to calculate the perturbative coefficients of the eigenenergies formal series, in the case of the Coulomb potential with a radial linear term and the radial quartic anharmonic oscillator potential. This calculation method, contrary to the usual Rayleigh-Schrodinger Perturbation Theory (RSPT), does not require the calculation of eigenfunctions coefficients. This method is a fast and efficient tool for the calculation of large order eigenenergies coefficients.
We study the problem of a particle hopping on the Bethe lattice in the presence of a Coulomb potential. We obtain an exact solution to the particle's Green's function along with the full energy spectrum. In addition, we present a mapping of a generalized radial potential problem defined on the Bethe lattice to an infinite number of one dimensional problems that are easily accessible numerically. The latter method is particularly useful when the problem admits no analytical solution.
We present a new exactly solvable quantum problem for which the Schroedinger equation allows for separation of variables in oblate spheroidal coordinates. Namely, this is the quantum mechanical two Coulomb centers problem for the case of imaginary intercenter parameter and complex conjugate charges is considered. Since the potential is defined by the two-sheeted mapping whose singularities are concentrated on a circle rather than at separate points, there arise additional possibilities in choice of boundary conditions. Detailed classification of the various types of boundary-value problems is …
It is shown that the radial Schroedinger equation for a power law potential and a particular angular momentum may be transformed using a change of variable into another Schroedinger equation for a different power law potential and a different angular momentum. It is shown that this leads to a mapping of the spectra of the two related power law potentials. It is shown that a similar correspondence between the classical orbits in the two related power law potentials exists. The well known correspondence of the Coulomb and oscillator spectra is a special case of a more general correspondence betw…
Schroedinger's equation with the attractive potential V(r) = -Z/(r^q+ b^q)^(1/q), Z > 0, b > 0, q >= 1, is shown, for general values of the parameters Z and b, to be reducible to the confluent Heun equation in the case q=1, and to the generalized Heun equation in case q=2. In a formulation with correct asymptotics, the eigenstates are specified a priori up to an unknown factor. In certain special cases this factor becomes a polynomial. The Asymptotic Iteration Method is used either to find the polynomial factor and the associated eigenvalue explicitly, or to construct accurate approximations f…
The regular solutions to the Schördinger equation in the case of an electron experiencing a Coulomb force are well known. Being that the radial part of the differential equation to be solved is second order in derivatives, it will have two independent solutions, with the second irregular solutions being ill-behaved at the origin and unbound at infinity. For this reason, second solutions are dropped for bound electrons. However, these second solutions are still of academic interest for several reasons. One reason is to help devise schemes to control numerical contamination by second solutions t…
We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For general radially symmetric potentials, we derive the asymptotic expansions of the log-partition functions up to and including the $O(1)$-terms as the number $N$ of particles increases. Notably, our findings stress that the formulas of the $O(\log N)$- and $O(1)$-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions prop…
For a competitive system of k coupled nonlinear Schroedinger equations we prove the existence, when the competition parameter is large, of positive radial solutions on R^N. We show that, when the competition parameter goes to infinity, the profile of each component separates, in many pulses, from the others. Moreover, we can prescribe the location of such pulses in terms of the oscillations of the changing-sign solutions of the scalar nonlinear Schroedinger equation. Within an Hartree-Fock approximation, this provides a theoretical indication of phase separation into many nodal domains for the…
With the aid of a modified similarity transformation we obtained exact energy eigenvalues of the generalized Dirac-Coulomb equation. This equation consists of the time component of the Lorentz 4-vector potential V_v(r)=-A_1/r, and a Lorentz scalar potential V_s(r)=-A_2/r. The transformed radial equations are so simple that their solutions are inferred from the conventional solutions of the Schrodinger equation.
We apply the Schrödinger factorization method to the radial second-order equation for the relativistic Kepler-Coulomb problem. From these operators we construct two sets of one-variable radial operators which are realizations for the $su(1,1)$ Lie algebra. We use this algebraic structure to obtain the energy spectrum and the supersymmetric ground state for this system.
We consider radial solutions to the Schrödinger-Poisson system in three dimensions with an external smooth potential with Coulomb-like decay. Such a system can be viewed as a model for the interaction of dark matter with a bright matter background in the non-relativistic limit. We find that there are infinitely many critical points of the Hamiltonian, subject to fixed mass, and that these bifurcate from solutions to the associated linear problem at zero mass. As a result, each branch has a different topological character defined by the number of zeros of the radial states. We construct numeric…
We decouple the Dirac's radial equations in $D+1$ dimensions with Coulomb-type scalar and vector potentials through appropriate transformations. We study each of these uncoupled second-order equations in an algebraic way by using an $su(1,1)$ algebra realization. Based on the theory of irreducible representations, we find the energy spectrum and the radial eigenfunctions. We construct the Perelomov coherent states for the Sturmian basis, which is the basis for the unitary irreducible representation of the $su(1,1)$ Lie algebra. The physical radial coherent states for our problem are obtained b…
The subject of this work are random Schroedinger operators on regular rooted tree graphs $\T$ with stochastically homogeneous disorder. The operators are of the form $H_λ(ω) = T + U + λV(ω)$ acting in $\ell^2(\T)$, with $T $ the adjacency matrix, $U$ a radially periodic potential, and $V(ω)$ a random potential. This includes the only class of homogeneously random operators for which it was proven that the spectrum of $H_λ(ω)$ exhibits an absolutely continuous (ac) component; a results established by A. Klein for weak disorder, in case U=0 and $V(ω)$ given by iid random variables on $\T$. Our m…
We present a generalization of the Fefferman-de la Llave decomposition of the Coulomb potential to quite arbitrary radial functions $V$ on $R^n$ going to zero at infinity. This generalized decomposition can be used to extend previous results on $N$-body quantum systems with Coulomb interaction to a more general class of interactions. As an example of such an application we derive the high density asymptotics of the ground state energy of jellium with Yukawa interaction in the thermodynamic limit, using a correlation estimate by Graf and Solovej.
In their recent paper (Inter. J. Mod. Phys. A 26 (2011) 1011), Zarrinkamar and coauthors have considered the radial Dirac equation for a Coulomb scalar, vector and tensor interaction. The exact solutions for the energy eigenvalues they have reported for spin-symmetry and pseudo-spin symmetry were mishandled and are incorrect eigenvalues, therefore.
The bound-state solutions and the su(1,1) description of the $d$-dimensional radial harmonic oscillator, the Morse and the $D$-dimensional radial Coulomb Schrödinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrum generating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schrödinger equations containing some position-dependent mass. The deformed su(1,1) construction recently achieved for a $d$-dimensional radial harmonic oscillator i…
Quantum mechanical scalar particle with polarizability is considered in the presence of the Coulomb field. Separation of variables is performed with the use of Wigner $D$-functions, the radial system of 15 equations is reduced to a single second order differential equation, which among the Coulomb term includes an additional interaction term of the form σα^{2} / M^{2}r^{4}. Various physical regimes exist that is demonstrated by examining the behavior of the curves of generalized squared radial momentum P^{2}(r). Eigenstates of the equations can be constructed in terms of double confluent Heun …
We consider in this note a class of two-dimensional determinantal Coulomb gases confined by a radial external field. As the number of particles tends to infinity, their empirical distribution tends to a probability measure supported in a centered ring of the complex plane. A quadratic confinement corresponds to the complex Ginibre Ensemble. In this case, it is also already known that the asymptotic fluctuation of the radial edge follows a Gumbel law. We establish in this note the universality of this edge behavior, beyond the quadratic case. The approach, inspired by earlier works of Kostlan a…
The Laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron-structure theory, with applications in hadronic and nuclear theory as well. While similar in form to the Coulomb bound-state eigenfunctions (from the Schroedinger eigenproblem) or the Coulomb-Sturmian functions (from a related Sturm-Liouville problem), the Laguerre functions, unlike these former functions, constitute a complete, discrete, orthonormal set for square-integrable functions in three dimensions. We construct the SU(1,1)xSO(3) dynamical algebra for the Laguerre functi…
New inverse and half-inverse problems: {\it sliding problems} are introduced. In this way several physically important equations are recovered from the quantum defect. In particular, sliding problems are solved for radial Schrödinger equation, radial Dirac system and multidimensional Schrödinger equation. Systems with Coulomb-type potentials are considered as well.
We use the tridiagonal representation approach to solve the radial Schrödinger equation for the continuum scattering states of the Coulomb problem in a complete basis set of discrete Bessel functions. Consequently, we obtain a new representation of the confluent hypergeometric function as an infinite sum of Bessel functions, which is numerically very stable and more rapidly convergent than another well-known formula.
Bound-state solutions of the singular harmonic oscillator and singular Coulomb potentials in arbitrary dimensions are generated in a simple way from the solutions of the one-dimensional generalized Morse potential. The nonsingular harmonic oscillator and nonsingular Coulomb potentials in arbitrary dimensions with their additional accidental degeneracies are obtained as particular cases. Added bonuses from these mappings are the straightforward determination of the critical attractive singular potential, the proper boundary condition on the radial eigenfunction at the origin and the inexistence…
We prove Strichartz estimates for the Schroedinger equation with an electromagnetic potential, in dimension $n\geq3$. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a non trapping condition, which are expressed as smallness of suitable components of the potentials. However, the potentials themselves can be large, and we avoid completely any a priori spectral assumption on the operator. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potent…
In this work we compute bound solutions for particles and anti-particles of the Dirac equation for a pure tensor radial Coulomb potential plus a constant. We find that the binding depends on the sign the tensor constant potential, and allows only bound solutions for a certain sign and magnitude of the $κ$ quantum number, which is related to the spin-orbit coupling in the Dirac equation. This relation is reversed for anti-particle solutions. On the other hand, the Coulomb tensor potential, although not biding, changes the range of $κ$ values for which there are bound solutions.
We regard the cosmological fluid within an exponentially expanding FLRW spacetime as the probability fluid of a nonrelativistic Schroedinger field. The scalar Schroedinger particle so described has a mass equal to the total (baryonic plus dark) matter content of the Universe. This procedure allows a description of the cosmological fluid by means of the operator formalism of nonrelativistic quantum theory. Under the assumption of radial symmetry, a quantum operator proportional to $1/r^2$ represents the cosmological constant $Λ$. The experimentally measured value of $Λ$ is one of the eigenvalue…
We report on our analysis of the Coulomb singularity problem in the frame of the coupled channel scattering theory including spin-orbit interaction. We assume that the coupling between the partial wave components involves orbital angular momenta such that $Δl = 0, \pm 2$. In these conditions, the two radial functions, components of a partial wave associated to two values of the angular momentum $l$, satisfy a system of two second-order ordinary differential equations. We examine the difficulties arising in the analysis of the behavior of the regular solutions near the origin because of this co…
We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac-Hardy inequality and exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate ou…
In this work, we obtain bound states for a nonrelativistic spin-half neutral particle under the influence of a Coulomb-like potential induced by the Lorentz symmetry breaking effects. We present a new possible scenario of studying the Lorentz symmetry breaking effects on a nonrelativistic quantum system defined by a fixed space-like vector field parallel to the radial direction interacting with a uniform magnetic field along the z-axis. Furthermore, we also discuss the influence of a Coulomb-like potential induced by Lorentz symmetry violation effects on the two-dimensional harmonic oscillator…
