On the order of levels in charmonium

We prove a theorem concerning the energies of the 2S and 3D states in a potential $\text{V(r) =}\text{?g}{}^2\text{r}\text{+ V}{}_{\mathrm{<mtext>c</mtext>}}\text{(r)}$, where V_c is a non-singular confining potential. If $\text{(}\text{d}\text{d}\text{r)}{}^3\text{(r}{}^2\text{V}{}_{\mathrm{<mtext>c</mtext>}}\text{)}$ is positive, then the 3D state lies above the 2S state, provided

$\text{d}\text{d}\text{r}\text{1}\text{r}\text{d}\text{d}\text{r}\text{2V}{}_{\mathrm{<mtext>c</mtext>}}\text{+r}\text{d}\text{V}{}_{\mathrm{<mtext>c</mtext>}}\text{d}\text{r}\text{< 0, ?}{}_{\mathrm{r}}\text{>0.}$

For V_c = r^α, this corresponds to 0 < α < 2.     

A new proof of the invariance principle in scattering theory

For free and interacting Hamiltonians, H₀ and H = H₀ + V(r) acting in L²(R³, dx) with V(r) a radial potential satisfying certain technical conditions, and for ? a real function on R with ?′ > 0 except on a discrete set, we prove that the Moller wave operators

exist and are independent of ?. The scattering operator

$\text{S = (Ω}{}^{\mathrm{+}}\text{)1Ω}{}^{\mathrm{?}}$

is shown to be unitary. Our proof utilizes time independent methods (eigenfunction expansions) and is effective in cases not previously analyzed, e.g. $\text{V(r) =}\text{sin}\text{r}\text{r}$ and many others.     

Two theorems on the level order in potential models

We study the potentials of the form U(r)=?r^?1+λV(r), $\text{(}\text{d}\text{d}\text{r}\text{)(}\text{r}{}^2\text{d}\text{V}\text{d}\text{r}\text{)?0}$, and show that the energy levels satisfy the inequalities $\text{E(N}{}_{\mathrm{c}}\text{, l)?E(N}{}_{\mathrm{c}}\text{, l+1)}$ to first order in λ, where N_c denotes the coulombic principal quantum number and l the angular momentum. Similarly for potentials $\text{U(r)=r}{}^2\text{+λV(r), (}\text{d}\text{d}\text{r}{}^2\text{)}{}^2\text{V(r)?0}$, we prove to first order in λ that $\text{E}\text{?}\text{(N}{}_{\mathrm{H}}\text{,}\text{l}\text{)?}\text{E}\text{?}\text{(N}{}_{\mathrm{H}}\text{,}\text{l}\text{+2)}$, where NH denotes the harmonic oscillator quantum number. In the latter case, we give also quantitative restrictions on the relative positions at the lth and (l+1)th states.     

A remarkable duality in one-particle quantum mechanics between some confining potentials and (R+L∞ϵ) potentials

The equivalence between the bound-state problem for the spatial harmonic oscillator and for the Coulomb potential is reexamined and extended: a similar duality is displayed between large classes of confining potentials and (R+L^∞_?) potentials, i.e., potentials for which the Schrödinger operator has among other properties a discrete spectrum bounded below, in (-∞, 0). The involved confining potentials U(r) must satisfy: $\text{r}{}^{\mathrm{-\alpha }}\text{U(r)}\textcolor{red}{}\text{g>0}{}_{\mathrm{r\to +\infty }}\text{, r}{}^{\mathrm{<mtext>-2\alpha </mtext><mtext>(2+\alpha )</mtext>}}\text{U (r}{}^{\mathrm{<mtext>2</mtext><mtext>(2+\alpha )</mtext>}}\text{) ? g}\text{is R}\text{+}\text{L}{}^{\mathrm{\infty }}{}_{\mathrm{?}}$.     

Energy levels of bound polarons

A new variational wave function to describe the ground state and the excited states of a bound polaron is proposed. It is of the form

$\text{|Ψ〉 = c|O〉|ø}{}_{\mathrm{n}}\text{〉 +}\text{∑}\text{g}{}_{\mathrm{k}}{}^1\text{V}{}_{\mathrm{k}}{}^1\text{(e}{}^{\mathrm{<mtext>i</mtext><mtext>k·r</mtext>}}\text{? ρ}{}_{\mathrm{k}}{}^1\text{)a}{}_{\mathrm{k}}{}^{\mathrm{+}}\text{|O〉|ø}{}_{\mathrm{n}}$

. It is argued that this form is reasonable for all electron—phonon coupling α and all strengths β of the Coulomb potential. Numerical and analytical results are derived for the energy of the ground state and compared to existing results. Results for the energy of the lowest p-type excited state of the bound polaron are obtained.     

Electron-hole interaction in the presence of excitons

We calculate the effective electron-hole interaction V_re in the presence of an exciton gas, which reads in real space:

$\text{V}{}_{\mathrm{re}}\text{(r)=}\text{?e}{}^2\text{r}\text{{1+}\text{∑}\text{i=1}\text{4}\text{(?1)}{}^{\mathrm{i}}\text{C}{}_{\mathrm{i}}\text{exp}\text{(?Z}{}_{\mathrm{i}}\text{r}\text{a}\text{}}$

The parameters C_i and Z_i are given explicitly for GaAs. For this material, we show the binding energy of the exciton is weakly modified so long as $\text{8πR}{}_0\text{?}{}_{\mathrm{ex}}\text{a}{}_0{}^3\text{kT?1}$. (R₀, exciton Rydberg, a₀ exciyon radius, ?_ex exciton density, T temperature).     

The energy of a system of relativistic massless bosons bound by oscillator pair potentials

We study the lowest energy E of a semirelativistic system of N identical massless bosons with Hamiltonian $\text{H=}\text{∑}\text{i=1}\text{N}\text{p}{}_{\mathrm{i}}{}^2\text{+}\text{∑}\text{j>i=1}\text{N}\text{γ|}\text{r}{}_{\mathrm{i}}\text{−}\text{r}{}_{\mathrm{j}}\text{|}{}^2\text{,}\text{γ>0.}$ We prove $\text{A}\text{γN}{}^2\text{(N−1)}{}^2{}^{\mathrm{1/3}}\text{⩽E⩽B}\text{γN}{}^2\text{(N−1)}{}^2{}^{\mathrm{1/3}}\text{,}$ where A=2.33810741 and $\text{B=}\text{81}\text{2π}{}^{\mathrm{1/3}}\text{=2.3447779}$. The average of these bounds determines E with an error less than 0.15% for all N⩾2.     

The ′T Hooft-Polyakov monopole near the Prasad-Sommerfield limit

A perturbative classical monopole solution for the SO(3) gauge theory is constructed in the limit of small but non-vanishing Higgs potential. This corresponds to the limit $\text{μ}{}^2\text{2M}{}_{\mathrm{W}}{}^2\text{=}\text{λ}\text{? 1}$, where μ equals the mass of the scalar particle and M_W equals the mass of the intermediate vector particles. The monopole solution and mass are found to involve non-analytic functions of λ: $\text{γ}$ and λ ln λ. The monopole mass M_m is calculated to order $\text{μ}{}^2\text{M}{}_{\mathrm{W}}$ as

$\text{M}{}_{\mathrm{m}}\text{=}\text{4π}\text{e}{}^2\text{M}{}_{\mathrm{w}}\text{1+}\text{1}\text{2}\text{μ}\text{M}{}_{\mathrm{w}}\text{+}\text{1}\text{2}\text{μ}{}^2\text{M}{}^2{}_{\mathrm{w}}\text{ln}\text{μ}\text{M}{}_{\mathrm{w}}\text{+0.7071}\text{μ}{}^2\text{M}{}^2{}_{\mathrm{w}}$

.     

Derivation of the classical theory of correlations in fluids by means of functional differentiation

The k-particle, infinite-volume distribution functions $\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ)}$ and modified Ursell correlation functions $\text{U}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ)}$ of a classical system of particles with the two-body potential $\text{q(}\text{r}\text{) + γ}{}^{\mathrm{\nu }}\text{K(γ}\text{r}\text{)}$ are considered. The limiting values of the functions $\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ),}\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{S}{}_1\text{/γ, …,}\text{S}{}_{\mathrm{k?1}}\text{/γ, γ)}$ and $\text{γ(}{}^{\mathrm{1?k}}\text{)ν}\text{U}\text{?}{}_{\mathrm{k}}\text{(}\text{S}{}_1\text{/γ, …,}\text{S}{}_{\mathrm{k?1}}\text{/γ, γ)}$ in the limit γ → 0 are calculated, under fairly weak conditions on q and K, by a method involving functional differentiation. These limiting functions are used to describe the molecular structure of the various states of the system both in the range of the potential q(r) and in the rage of the potential γ^νK(γr). The direct correlation function c? (r, γ) is also considered and it is shown that for $\text{S}\text{≠ 0}$, $\text{lim}{}_{\mathrm{\gamma \to 0}}\text{γ}{}^{\mathrm{?\nu }}\text{c}\text{?}\text{(}\text{S}\text{γ}\text{, γ) = ?βK (}\text{S}\text{)}$, for all one-phase states, where β is the reciprocal temperature. Special cases of our results confirm those of other authors, including the well-known results of Ornstein and Zernike.     

Inverse solutions to the inelastic scattering problem

Exact inverse solutions to the integral equation $\text{φ(r}{}_{\mathrm{s}}\text{|r}{}_0\text{, k) = ?}{}_{\mathrm{D}}{}^3\text{f (r, ω)g(r|r}{}_0\text{, k)g(r|r, k)d}{}^3\text{r}$, where g(r|r_j, k); j = 0 or s is the free space Green function, are derived in plane and cylindrical coordinates for fixed ω. These solutions allow an inelastic scattering potential f(r, ω) which is of compact support r ? $\text{D}$³ to be recovered from scattering data collected over the surfaces of a plane and cylinder respectively.     

The emission spectrum of SeO in the far ultraviolet

The emission spectrum of SeO in the far ultraviolet first observed by Haranath (1) at low dispersion has been photographed in the region 2480-1930 Å under medium resolution and a reanalysis of the vibrational structure of the bands has been presented. Beginning at the longer wavelength end, the spectrum has been analyzed into five band systems which are designated as $\text{c(}{}^1\text{Σ}{}^{\mathrm{+}}\text{)-b(}{}^1\text{Σ}{}^{\mathrm{+}}\text{)}$, x₂-x₁, y₂-y₁, $\text{C(}{}^3\text{Π}\text{)-X}{}^3\text{Σ}{}^{\mathrm{?}}$, and $\text{D(}{}^3\text{Σ}{}^{\mathrm{?}}\text{)-X}{}^3\text{Σ}{}^{\mathrm{?}}$. The lower state of the c-b system is found to be the upper state of the $\text{b(}{}^1\text{Σ}{}^{\mathrm{+}}\text{)-X}{}^3\text{Σ}{}^{\mathrm{?}}$ system observed recently by us (2). The derived constants in cm^?1 for SeO are as follows (the constants of the b state are those derived from Ref. 2).

     

State	$\text{T}{}_{\mathrm{e}}$	$\text{ω}{}_{\mathrm{e}}$	$\text{ω}{}_{\mathrm{e}}\text{x}{}_{\mathrm{e}}$	λ
$\text{y}{}_2$	$\text{y}{}_1\text{+ 47830}$	974	6.0
$\text{y}{}_1$	$\text{y}{}_1$	906	21.0
$\text{x}{}_2$	$\text{x}{}_1\text{+ 46009}$	993	2.0
$\text{x}{}_1$	$\text{x}{}_1$	877	8.0
$\text{c(}{}^1\text{Σ}{}^{\mathrm{+}}\text{)}$	53080	954	13.0
$\text{D(}{}^3\text{Σ}{}^{\mathrm{?}}\text{)}\text{F}{}_2$	51422	955	9.3
F1	51356	955	8.5	～36
$\text{C(}{}^3\text{Π}\text{)}$	50873	1034	9.3
$\text{b(}{}^1\text{Σ}{}^{\mathrm{+}}\text{)}$	9570.7	834.9	5.5