For any central potential V in D dimensions, the angular Schroedinger equation remains the same and defines the so called hyperspherical harmonics. For non-central models, the situation is more complicated. We contemplate two examples in the plane: (1) the partial differential Calogero's three-body model (without centre of mass and with an impenetrable core in the two-body interaction), and (2) the Smorodinsky-Winternitz' superintegrable harmonic oscillator (with one or two impenetrable barriers). These examples are solvable due to the presence of the barriers. We contemplate a small complex shift of the angle. This creates a problem: the barriers become "translucent" and the angular potentials cease to be solvable, having the sextuple-well form for Calogero model and the quadruple or double well form otherwise. We mimic the effect of these potentials on the spectrum by the multiple, purely imaginary square wells and tabulate and discuss the result in the first nontrivial double-well case.
The intertwining operator technique is applied to difference Schroedinger equations with operator-valued coefficients. It is shown that these equations appear naturally when a discrete basis is used for solving a multichannel Schroedinger equation. New families of exactly solvable multichannel Hamiltonians are found.
We propose a general method for constructing quasi-exactly solvable potentials with three analytic eigenstates. These potentials can be real or complex functions but the spectrum is real. A comparison with other methods is also performed.
We construct a general algorithm generating the analytic eigenfunctions as well as eigenvalues of one-dimensional stationary Schroedinger Hamiltonians. Both exact and quasi-exact Hamiltonians enter our formalism but we focus on quasi-exact interactions for which no such general approach has been considered before. In particular we concentrate on a generalized sextic oscillator but also on the Lame and the screened Coulomb potentials.
New examples of matrix quasi exactly solvable Schroedinger operators are constructed. One of them constitutes a matrix generalization of the quasi exactly solvable anharmonic oscillator, the corresponding invariant vector space is constructed explicitely. Also investigated are matrix generalizations of the Lame equation.
Two families of quasi exactly solvable 2*2 matrix Schroedinger operators are constructed. The first one is based on a polynomial matrix potential and depends on three parameters. The second is a one-parameter generalisation of the scalar Lame equation. The relationship between these operators and QES Hamiltonians already considered in the literature is pointed out.
A new class of quasi exactly solvable potentials with a variable mass in the Schroedinger equation is presented. We have derived a general expression for the potentials also including Natanzon confluent potentials. The general solution of the Schroedinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials.
Some exactly solvable potentials in the position dependent mass background are generated whose bound states are given in terms of Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These potentials are shown to be shape invariant and isospectral to the potentials whose bound state solutions involve classical Laguerre or Jacobi polynomials.
The stationary Schroedinger equation of the harmonic oscillator is deformed by a Darboux transformation to construct time-dependent potentials with the oscillator profile. The Darboux (supersymmetric or factorization) method is usually developed in the spatial variables of the Schroedinger equation. Here we follow a variation introduced by Bagrov, Samsonov and Shekoyan to include the time-variable as a parameter of the transformation.
A one-dimensional Schrödinger equation with position-dependent effective mass in the kinetic energy operator is studied in the framework of an $so(2,1)$ algebra. New mass-deformed versions of Scarf II, Morse and generalized Pöschl-Teller potentials are obtained. Consistency with an intertwining condition is pointed out.
An exactly solvable position-dependent mass Schrödinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical w…
A new family of solvable potentials related to the Schroedinger-Riccati equation has been investigated. This one-dimensional potential family depends on parameters and is restricted to the real interval. It is shown that this potential class, which is a rather general class of solvable potentials related to the hypergeometric functions, can be generalized to even wider classes of solvable potentials. As a consequence the nonlinear Schroedinger-type equation has been obtained.
An algebraic method of constructing potentials for which the Schroedinger equation with position dependent mass can be solved exactly is presented. A general form of the generators of su(1,1) algebra has been employed with a unified approach to the problem. Our systematic approach reproduces a number of earlier results and also leads to some novelties. We show that the solutions of the Schroedinger equation with position dependent mass are free from the choice of parameters for position dependent mass. Two classes of potentials are constructed that include almost all exactly solvable potential…
At the lower edge of the energy continuum the birth of an isolated quantum bound state is studied as caused by an infinitesimally small change of the interaction. In our model a single, asymptotically free massive quantum particle is assumed moving along a discretized real line of coordinates, $x \in \mathbb{Z}$. The motion is assumed controlled by a weakly nonlocal 2J-parametric external potential which is non-Hermitian but PT-symmetric. Mathematically, the bound states are then reinterpreted as Sturmians, i.e., the bound-state energy is treated as a variable real parameter while the value of…
We construct six multi-parameter families of Hermitian quasi-exactly solvable matrix Schroedinger operators in one variable. The method for finding these operators relies heavily upon a special representation of the Lie algebra o(2,2) whose representation space contains an invariant finite-dimensional subspace. Besides that we give several examples of quasi-exactly solvable matrix models that have square-integrable eigenfunctions. These examples are in direct analogy with the quasi-exactly solvable scalar Schroedinger operators obtained by Turbiner and Ushveridze.
We construct arbitrary-order Darboux transformations for Schroedinger equations with energy-dependent potential and position-dependent mass within the Dunkl formalism. Our construction is based on a point transformation that interrelates our equations with the standard Schroedinger case. We apply our method to generate several solvable Dunkl-Schroedinger equations.
Angular parts of certain solvable models are studied. We find that an extension of this class may be based on suitable trigonometric identities. The new exactly solvable Hamiltonians are shown to describe interesting two- and three-particle systems of the generalized Calogero, Wolfes and Winternitz-Smorodinsky types.
A new approach in solution of simple quantum mechanical problems in deformed space with minimal length is presented. We propose the generalization of Schroëdinger equation in momentum representation on the case of deformed Heisenberg algebra with minimal length. Assuming that the kernel of potential energy operator do not change in the case of deformation, we obtain exact solution of eigenproblem of a particle in delta potential as well as double delta potential. Particle in Coulomb like potential is revisited and the problem of inversibility and hermicity of inverse coordinate operator is sol…
Using an isomorphism between Hilbert spaces $L^2$ and $\ell^{2}$ we consider Hamiltonians which have tridiagonal matrix representations (Jacobi matrices) in a discrete basis and an eigenvalue problem is reduced to solving a three term difference equation. Technique of intertwining operators is applied to creating new families of exactly solvable Jacobi matrices. It is shown that any thus obtained Jacobi matrix gives rise to a new exactly solvable non-local potential of the Schroedinger equation. We also show that the algebraic structure underlying our approach corresponds to supersymmetry. Sup…
A new convenient method to diagonalize the non-relativistic many-body Schroedinger equation with two-body central potentials is derived. It combines kinematic rotations (democracy transformations) and exact calculation of overlap integrals between bases with different sets of mass-scaled Jacobi coordinates, thereby allowing for a great simplification of this formidable problem. We validate our method by obtaining a perfect correspondence with the exactly solvable three-body ($N=3$) Calogero model in 1D.
A comprehensive review of exactly solvable quantum mechanics is presented with the emphasis of the recently discovered multi-indexed orthogonal polynomials. The main subjects to be discussed are the factorised Hamiltonians, the general structure of the solution spaces of the Schroedinger equation (Crum's theorem and its modifications), the shape invariance, the exact solvability in the Schroedinger picture as well as in the Heisenberg picture, the creation/annihilation operators and the dynamical symmetry algebras, coherent states, various deformation schemes (multiple Darboux transformations)…
Based on the well-known relation between Fokker-Planck equations and Schroedinger equations of quantum mechanics (QM), we propose new deformed Fokker-Planck (FP) equations associated with the Schroedinger equations of "discrete" QM. The latter is a natural discretization of QM and its Schroedinger equations are difference instead of differential equations. Exactly solvable FP equations are obtained corresponding to exactly solvable "discrete" QM, whose eigenfunctions include various deformations of the classical orthogonal polynomials.
Three-body Schroedinger equation is studied in one dimension. Its two-body interactions are assumed composed of the long-range attraction (dominated by the L-th-power potential) in superposition with a short-range repulsion (dominated by the (-K)-th-power core) plus further subdominant power-law components if necessary. This unsolvable and non-separable generalization of Calogero model (which is a separable and solvable exception at L = K = 2) is presented in polar Jacobi coordinates. We derive a set of trigonometric identities for the potentials which generalizes the well known K=2 identity o…
It is proved that when solving Schroedinger equations for radially symmetric potentials the effect of higher dimensions on the radial wave function is equivalent to the effect of higher angular momenta in lower dimensional cases. This result is applied to giving solutions for several radially symmetric potentials in N-dimension.
We construct energy-dependent potentials for which the Schroedinger equations admit solu- tions in terms of exceptional orthogonal polynomials. Our method of construction is based on certain point transformations, applied to the equations of exceptional Hermite, Jacobi and Laguerre polynomials. We present several examples of boundary-value problems with energy-dependent potentials that admit a discrete spectrum and the corresponding normalizable solutions in closed form.
We present an explicit convergent iterative solution for the lowest energy state of the Schroedinger equation with an $N$-dimensional radial potential $V=\frac{g^2}{2}(r^2-1)^2$ and an angular momentum $l$. For $g$ large, the rate of convergence is similar to a power series in $g^{-1}$.
A new type of supersymmetric transformations of the coupled-channel radial Schroedinger equation is introduced, which do not conserve the vanishing behavior of solutions at the origin. Contrary to usual transformations, these ``non-conservative'' transformations allow, in the presence of thresholds, the construction of potentials with coupled scattering matrices from uncoupled potentials. As an example, an exactly-solvable potential matrix is obtained which provides a very simple model of Feshbach-resonance phenomenon.
The scattering state solutions of the Klein-Gordon equation with equal scalar and vector Varshni, Hellmann and Varshni-Shukla potentials for any arbitrary angular momentum quantum number l are investigated within the framework of the functional analytical method using a suitable approximation. The asymptotic wave functions, approximate scattering phase shifts, normalization constants and bound state energy equations were obtained. The non-relativistic limits of the scattering phase shifts and the bound states energy equations for the three potentials were also obtained. Our bound states energy…
We present a procedure to solve the Schroedinger equation of two interacting electrons in a quantum dot in the presence of an external magnetic field within the context of quasi-exactly-solvable spectral problems. We show that the symmetries of the Hamiltonian can be recovered for specific values of the magnetic field, which leads to an exact determination of the eigenvalues and eigenfunctions. We show that the problem possesses a hidden sl_2-algebraic structure.
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…
Analytical solutions of the Klein-Gordon equation are obtained by reducing the radial part of the wave equation to a standard form of a second order differential equation. Differential equations of this standard form are solvable in terms of hypergeometric functions and we give an algebraic formulation for the bound state wave functions and for the energy eigenvalues. This formulation is applied for the solutions of the Klein-Gordon equation with some diatomic potentials.
We consider a charged particle driven by a time-dependent flux threading a quantum ring. The dynamics of the charged particle is investigated using classical treatment, Fourier expansion technique, time-evolution method, and Lewis-Riesenfeld approach. We have shown that, by properly managing the boundary conditions, a time-dependent wave function can be obtained using a general non-Hermitian time-dependent invariant, which is a specific linear combination of initial angular-momentum and azimuthal-angle operators. It is shown that the linear invariant eigenfunction can be realized as a Gaussian…
We solve an eigenvalue equation that appears in several papers about a wide range of physical problems. The Frobenius method leads to a three-term recurrence relation for the coefficients of the power series that, under suitable truncation, yields exact analytical eigenvalues and eigenfunctions for particular values of a model parameter. From these solutions some researchers have derived a variety of predictions like allowed angular frequencies, allowed field intensities and the like. We also solve the eigenvalue equation numerically by means of the variational Rayleigh-Ritz method and compare…
Orthogonal Polynomials in Quantum Mechanics. Exact solutions of the Schrodinger equation with the hyperbolic Scarf potential (Scarf II) in terms of Romanovski polynomials. Among the applications included is the solution of the problem of an electron within a particular potential leading to the definition of new non-spherical angular functions.
Exactly solvable model of two Rydberg-dressed atoms moving in a quasi-two-dimensional harmonic trap is introduced and its properties are investigated. Depending on the strength of inter-particle interactions and the critical range of the potential, the two-particle eigenstates are classified with respect to the excitations of the center-of-mass motion, relative angular momentum, and relative distance variable. Having these solutions in hand, we discuss inter-particle correlations as functions of interaction parameters. We also present a straightforward prescription of how to generalize obtaine…
We present a supersymmetric analysis for the d-dimensional Schroedinger equation with the generalized isotonic nonlinear-oscillator potential V(r)={B^2}/{r^{2}}+ω^{2} r^{2}+2g{(r^{2}-a^{2})}/{(r^{2}+a^{2})^{2}}, B\geq 0. We show that the eigenequation for this potential is exactly solvable provided g=2 and (ωa^2)^2 = B^2 +(\ell +(d-2)/2)^2. Under these conditions, we obtain explicit formulae for all the energies and normalized bound-state wavefunctions.
A non-unitary version of quantum scattering is studied via an exactly solvable toy model. The model is merely asymptotically local since the smooth path of the coordinate is admitted complex in the non-asymptotic domain. At any real angular-momentum-like parameter the reflection R and transmission T are shown to change with the winding number (i.e., topology) of the path. The points of unitarity appear related to the points of existence of quantum-knot bound states.
We classify (1+3)-dimensional Pauli equations for a spin-1/2 particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the eleven classes of vector-potentials of the electro-magnetic field A(t,x) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is its equivalence to the system of two uncoupled Schroedinger equations. In addition, the magnetic field…
Two rectangular models described by the one-dimensional Schroedinger equation with sharply localized potentials are suggested. The potentials have a multi-layer thin structure being composed from adjacent barriers and wells. Their peculiar tunneling properties are studied in considerable detail. Particularly, in the zero-range limit when the potentials are squeezed to a single point, sharp peaks with total transmission are observed at certain (positive and negative) quantized values of the potential strength constant forming infinite discrete sets. Beyond these sets, the barrier-well structure…
Eigenvalues and eigenfunctions of the volume operator, associated with the symmetric coupling of three SU(2) angular momentum operators, can be analyzed on the basis of a discrete Schroedinger-like equation which provides a semiclassical Hamiltonian picture of the evolution of a `quantum of space', as shown by the authors in a recent paper. Emphasis is given here to the formalization in terms of a quadratic symmetry algebra and its automorphism group. This view is related to the Askey scheme, the hierarchical structure which includes all hypergeometric polynomials of one (discrete or continuou…
This is the fourth article in a series where we succeed in enlarging the class of exactly solvable quantum systems. We do that by working in a complete set of square integrable basis that carries a tridiagonal matrix representation for the wave operator. Consequently, the matrix wave equation becomes a three-term recursion relation for the expansion coefficients of the wavefunction. Finding solutions of this recursion relation in terms of orthogonal polynomials is equivalent to solving the original problem. This method, called the Tridiagonal Representation Approach (TRA), gives a larger class…
This is the second article in a series where we succeed in enlarging the class of solvable problems in one and three dimensions. We do that by working in a complete square integrable basis that carries a tridiagonal matrix representation of the wave operator. Consequently, the wave equation becomes equivalent to a three-term recursion relation for the expansion coefficients of the wavefunction in the basis. Finding solutions of the recursion relation is equivalent to solving the original problem. This method gives a larger class of solvable potentials. The usual diagonal representation constra…
We show how it is possible to trap two-dimensional massless Dirac fermions in spatially inhomogeneous magnetic fields, as long as the formed magnetic quantum dot (or ring) is of a slowly decaying nature. It is found that a modulation of the depth of the magnetic quantum dot leads to successive confinement-deconfinement transitions of vortexlike states with a certain angular momentum, until a regime is reached where only states with one sign of angular momentum are supported. We illustrate these characteristics with both exact solutions and a hitherto unknown quasi-exactly solvable model utiliz…
Quantum-mechanical wave functions in singular potentials: linear and nonlinear states
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University（九州大学理工学研究科）；Department of Physical Electronics, School of Electrical and Computer Engineering, Faculty of Engineering, Tel Aviv University（特拉维夫大学工程学院电子物理系）；Instituto de Alta Investigación, Universidad de Tarapacá（塔拉帕卡大学高级研究所）
This paper is concerned with the discrete spectra of Schroedinger operators H = -Delta + V, where V(r) is an attractive potential in N spatial dimensions. Two principal results are reported for the bottom of the spectrum of H in each angular-momentum subspace H_{ell}: (i) an optimized lower bound when the potential is a sum of terms V(r) = V^{(1)}(r) + V^{(2)}(r), and the bottoms of the spectra of -Delta + V^{(1)}(r) and -Delta + V^{(2)}(r) in H_{ell} are known, and (ii) a generalized comparison theorem which predicts spectral ordering when the graphs of the comparison potentials V^{(1)}(r) an…
