A single particle moves in one dimension and obeys nonrelativistic quantum mechanics. If the potential shape f(x) is symmetric and vanishes at infinity, then, without loss of generality, it can be represented in the form of a mixture f(x)=∫0∞A(t)w(x/t)dt of square wells w(x). It is shown that A(t)=f′(t) and that the discrete eigenvalues of the Schrödinger operator H=−D2+vf(x) are given approximately by the general formula En≂EAn=mins≳0[s+v∫0∞f′(t)w̄n(st2)dt], where {w̄n(s)}n=0∞ are the known ‘‘kinetic potentials’’ associated with the simple square-well problem. For n=0 or 1, the approximations EnA are lower bounds to the corresponding exact energies En. The general formula is applied to the sech-squared potential and to the truncated quartic oscillator.
A particle moves in one spatial dimension in an attractive symmetric negative potential vf(x), and obeys non-relativistic quantum mechanics. If the system is enclosed in a finite box then, unlike the situation in , the ground-state energy is negative only when the coupling v is sufficiently large. General upper- and lower-bound formulae are derived for the critical coupling which corresponds to . Some generalizations are derived for potentials which change sign and for unsymmetric unimodal potentials.
This paper concerns the ground-state energy
EN of a system of
N identical bosons interacting via the attractive central pair potential $V({r}_{\mathrm{ij}})=\ensuremath{-}{V}_{0}f(\frac{{r}_{\mathrm{ij}}}{a})$ and obeying nonrelativistic quantum mechanics. It is assumed that the potential shape
f is decreasing and can be represented as the envelope of each of two complementary families of power-law potentials $\ensuremath{\alpha}+\ensuremath{\beta}{r}^{p}$ (one family is above
f and the other below) for suitable fixed
p=p1 and
p=p2. If $\ensuremath{\epsilon}=\ensuremath{-}\frac{m{a}^{2}{E}_{N}}{(N\ensuremath{-}1){\ensuremath{\hbar}}^{2}}$ and $v=\frac{\mathrm{Nm}{V}_{0}{a}^{2}}{2{\ensuremath{\hbar}}^{2}}$, then it is proved that the entire collection of nonintersecting energy trajectories $\ensuremath{\epsilon}={F}_{N}(v)$,
N=2,3,4,…, is bounded between the fixed curves $(v, \ensuremath{\epsilon})=(\ensuremath{-}\ensuremath{\gamma}(p){[{s}^{3}{f}^{\ensuremath{'}}(s)]}^{\ensuremath{-}1}, (\frac{v}{2})[2f(s)+s{f}^{\ensuremath{'}}(s)])$, where the curve parameter
s>0, and
p=p1,p2. Potentials, for example, with shapes $f(r)=\frac{{\ensuremath{\alpha}}_{1}}{r}+\frac{{\ensuremath{\alpha}}_{2}}{(r+{\ensuremath{\alpha}}_{3})}\ensuremath{-}{\ensuremath{\alpha}}_{4}\mathrm{ln}r\ensuremath{-}{\ensuremath{\alpha}}_{5}\mathrm{sgn}(q){r}^{q}$, where ${\ensuremath{\alpha}}_{i}\ensuremath{\ge}0$ and $|q|\ensuremath{\le}1$, have the $\ensuremath{\gamma}$ numbers $\ensuremath{\gamma}(\ensuremath{-}1)=2$ and $\ensuremath{\gamma}(1)=\frac{12}{\ensuremath{\pi}}$. The appropriate $\ensuremath{\gamma}$ numbers are provided for other classes of potential shape including perturbed harmonic oscillators, and also for problems in one spatial dimension. The method yields in effect a recipe for the way
EN depends on
N and all the parameters of the pair potential.
We formulate a simple condition for reconstructibility of a certain class of Hamiltonians with real potentials from the knowledge of their complex-valued eigenfunctions. This may be relevant to the question of preparability of quantum states raised by W. Lamb in his 1969 paper on operational interpretation of quantum mechanics. Of particular interest to engineering applications are: (a) an exotic case of an upside-down harmonic-oscillator-type potential (a variation of the inverted "Mexican hat" potential) whose square-integrable complex eigenfunction describes a localized scattering state similar to the "bound state in the continuum" of von Neumann and Wigner, and (b) a spatially confined scattering state of a particle moving in an infinite well with the properly shaped potential bottom.
A nonrelativistic quantum mechanical particle moving freely on a curved surface feels the effect of the nontrivial geometry of the surface through the kinetic part of the Hamiltonian, which is proportional to the Laplace-Beltrami operator, and a geometric potential, which is a linear combination of the mean and Gaussian curvatures of the surface. The coefficients of these terms cannot be uniquely determined by general principles of quantum mechanics but enter the calculation of various physical quantities. We examine their contribution to the geometric scattering of a scalar particle moving on an asymptotically flat embedded surface. In particular, having in mind the possibility of an experimental realization of the geometric scattering in a low-density electron gas formed on a bumped surface, we determine the scattering amplitude for arbitrary choices of the curvature coefficients for a surface with global or local cylindrical symmetry. We also examine the effect of perturbations that violate this symmetry and consider surfaces involving bumps that form a lattice.
We present a new procedure to construct the one-dimensional non-Hermitian imaginary potential with a real energy spectrum in the context of the position-dependent effective mass Dirac equation with the vector-coupling scheme in 1+1 dimensions. In the first example, we consider a case for which the mass distribution combines linear and inversely linear forms, the Dirac problem with a PT -symmetric potential is mapped into the exactly solvable Schrödinger-like equation problem with the isotonic oscillator by using the local scaling of the wavefunction. In the second example, we take a mass distribution with smooth step shape, the Dirac problem with a non- PT -symmetric imaginary potential is mapped into the exactly solvable Schrödinger-like equation problem with the Rosen–Morse potential. The real relativistic energy levels and corresponding wavefunctions for the bound states are obtained in terms of the supersymmetric quantum mechanics approach and the function analysis method.