Symmetric Morse potential is exactly solvable

Abstract

Morse potential $V_M(x)=g^2\exp (2x)-g(2h+1)\exp (x)$ is defined on the full line, $-\infty <x<\infty$ and it defines an exactly solvable 1-d quantum mechanical system with finitely many discrete eigenstates. By taking its right half $0\le x<\infty$ and glueing it with the left half of its mirror image $V_M(-x)$, $-\infty <x\le 0$, the symmetric Morse potential $V(x)=g^2\exp (2|x|)-g(2h+1)\exp (|x|)$ is obtained. The quantum mechanical system of this piecewise analytic potential has infinitely many discrete eigenstates with the corresponding eigenfunctions given by the Whittaker W function. The eigenvalues are the square of the zeros of the Whittaker function $W_{k,\nu }(x)$ and its linear combination with $W_{k,\nu }^{\prime }(x)$ as a function of $\nu$ with fixed $k$ and $x$. This quantum mechanical system seems to offer an interesting example for discussing the Hilbert-Pólya conjecture on the pure imaginary zeros of Riemann zeta function on Re$(s)=\frac{1}{2}$.

LaTeX 14 pages, no figure, typos corrected, 1 reference added

References in corpus (4)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Orthogonal Polynomials from Hermitian Matrices
• Exactly solvable `discrete' quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation operators and coherent states
• Confining non-analytic exponential potential $V(x)=g^2\exp (2|x|)$ and its exact Bessel-function solvability