Exact solution of the Schrödinger equation for the inverse square root potential $V_0/\sqrt{x}$

Abstract

We present the exact solution of the stationary Schrödinger equation equation for the potential $V=V_0/\sqrt{x}$. Each of the two fundamental solutions that compose the general solution of the problem is given by a combination with non-constant coefficients of two confluent hypergeometric functions of a shifted argument. Alternatively, the solution is written through the first derivative of a tri-confluent Heun function. Apart from the quasi-polynomial solutions provided by the energy specification $E_n=E_1{n}^{-2/3}$, we discuss the bound-state wave functions vanishing both at infinity and in the origin. The exact spectrum equation involves two Hermite functions of non-integer order which are not polynomials. An accurate approximation for the spectrum providing a relative error less than $10^{-3}$ is $E_n=E_1(n-1/(2\pi ){)}^{-2/3}$ . Each of the wave functions of bound states in general involves a combination with non-constant coefficients of two confluent hypergeometric and two non-integer order Hermite functions of a scaled and shifted coordinate.

References in corpus (2)

• Thirty five classes of solutions of the quantum time-dependent two-state problem in terms of the general Heun functions
• Potentials of the Heun class: the triconfluent case

Cited by in corpus (36)

• Solutions of the bi-confluent Heun equation in terms of the Hermite functions
• Schrödinger potentials solvable in terms of the general Heun functions
• Discretization of Natanzon potentials
• Exact solution of the Schrödinger equation for a short-range exponential potential with inverse square root singularity
• The Lambert-W step-potential - an exactly solvable confluent hypergeometric potential
• A conditionally exactly solvable generalization of the inverse square root potential
• The third exactly solvable hypergeometric quantum-mechanical potential
• Exact solution of inverse-square-root potential $V\left(r\right)=-\frac{\alpha }{\sqrt{r}}$
• Exact solutions of the sextic oscillator from the bi-confluent Heun equation
• Maslov index for power-law potentials
• One-dimensional Schrödinger equation with non-analytic potential $V(x)=-g^2\exp (-|x|)$ and its exact Bessel-function solvability
• A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions
• Morse potential, symmetric Morse potential and bracketed bound-state energies
• Non-interacting fermions in hard-edge potentials
• Four five-parametric and five four-parametric independent confluent Heun potentials for the stationary Klein-Gordon equation
• Symmetrized quartic polynomial oscillators and their partial exact solvability
• Hermite function solutions of the Schrödinger equation for the sextic oscillator
• Two- and three-body problem with Floquet-driven zero-range interactions
• Exact solution of the 1D Dirac equation for the inverse-square-root potential $1/\sqrt{x}$
• Symmetrized exponential oscillator
• A conditionally integrable bi-confluent Heun potential involving inverse square root and centrifugal barrier terms
• Harmonic Oscillator with a Step and its Isospectral Properties
• Two new integrable cases of two-dimensional quantum mechanics with a magnetic field
• Two-dimensional Dirac fermion in presence of an asymmetric vector potential
• Exact surface-wave spectrum of a dilute quantum liquid
• Understanding Floquet Resonances in Ultracold Gas Scattering
• Separation equations for 2D superintegrable systems on constant curvature spaces
• A conditionally integrable Schrödinger potential of a bi-confluent Heun class
• Semiclassical rates for tunnel ionization from power-law potentials induced by a constant or low-frequency electric field
• Comment on the paper "Third-harmonic generation investigated by a short-range bottomless exponential potential well" by M. Hu, K. Guo, Q. Yu, Z. Zhang [Superlattices and Microstructures, 122 (2018) 538-547]
• Bi-confluent Heun potentials for a stationary relativistic wave equation for a spinless particle
• A new exactly integrable hypergeometric potential for the Schrödinger equation
• The third five-parametric hypergeometric quantum-mechanical potential
• On the Form of Solutions of Fuchsian differential Equations with n regular singular Points
• On Polynomial Solutions of Linear Differential Equations with Applications
• Power law duality in classical and quantum mechanics