\section{Our Battlefield: Semirelativistic Bound States of Least
Complexity}A popular (possibly even analytic) approach to
semirelativistic bound states of scalar particles is known under
the name spinless Salpeter equation, which is just the eigenvalue
equation of a Hamiltonian $H$ encompassing both the relativistic
kinetic term of the bound-state constituents and a static
potential $V$ encoding all interactions. For the case of two
bound-state constituents, of masses $m_1$ and $m_2$, and relative
coordinates and momenta $\bm{x}$ and $\bm{p}$, this Hamiltonian
$H$~reads\begin{equation}H=\sqrt{\bm{p}^2+m_1^2}+
\sqrt{\bm{p}^2+m_2^2}+V(\bm{x})\ ,\label{H}\end{equation}which,
for equal masses $m_1=m_2=m$ of its bound-state constituents,
simplifies (a little~bit)~to$$\widetilde H=2\,\sqrt{\bm{p}^2+m^2}+
V(\bm{x})\ .$$The spinless Salpeter equation may be regarded as
some approximation defined by a sequence of self-evident
assumptions that serves to simplify the homogeneous
Bethe--Salpeter equation \cite{BS}, which constitutes within the
framework of relativistic quantum field theories the adequate
approach to bound states of the fundamental degrees of freedom of
such quantum field theory:\begin{enumerate}
\item The most decisive simplification is to disregard any
reference to \emph{timelike\/} coordinate and momentum variables,
thereby defining an \emph{instantaneous\/} Bethe--Salpeter
formalism \cite{WLe}.\item Enabling any bound-state constituent to
propagate freely entails Salpeter's equation~\cite{SE}.\item
Skipping impacts of negative-energy contributions gives the
\emph{reduced\/} Salpeter equation.\item Ignoring the spins of all
bound-state constituents leads to the \emph{spinless\/} Salpeter
equation.\end{enumerate}

\section{General Insights: Elementary Constraints on Bound-State
Spectra}\label{S}Even if being not accessible analytically, the
spectrum of the operator $\widetilde H$ is tightly
constrained.\pagebreak
\begin{description}\item[1.~Boundedness from below of Hamiltonian
operators] is, as kinetic operators are positive, obvious if
$V(\bm{x})$ itself is bounded from below. If not, comparison with
the relativistic Coulomb problem may prove advantageous. The
Hamiltonian operator $\widetilde H$ with the Coulomb interaction
$$V(\bm{x})\equiv-\frac{\alpha}{|\bm{x}|}\ ,\qquad\alpha>0\ ,$$
will be bounded from below \cite{IWH1,IWH2} if and only if the
coupling strength $\alpha$ satisfies the constraint$$\alpha\le
\frac{4}{\pi}=1.273239\dots\ .$$Within this interval, the
spectrum, $\sigma(\widetilde H)$, of this operator $\widetilde H$
is subject to the lower bound~\cite{IWH1,IWH2}\begin{equation}
\sigma(\widetilde H)\ge2\,m\,\sqrt{1-\left(\frac{\pi\,\alpha}{4}
\right)^{\!2}}\ .\label{IWH}\end{equation}For Coulomb couplings
$\alpha\le1$, the lower bound (\ref{IWH}) may, straightforwardly,
be improved~\cite{MR}~to$$\sigma(\widetilde H)\ge2\,m\,
\sqrt{\frac{1+\sqrt{1-\alpha^2}}{2}}\ .$$\item[2.~Number of
discrete eigenstates in the spectrum of $\widetilde H$:]For all
potentials $V(\bm{x})$ that satisfy\begin{equation}V(\bm{x})=V(r)
\in L^{3/2}({\mathbb R}^3)\cap L^3({\mathbb R}^3)\ ,\qquad
V(\bm{x})\leq0\ ,\qquad r\equiv|\bm{x}|\ ,\label{L}\end{equation}
the number of bound states of such operator $\widetilde H$ may be
shown to be bounded from above~by~\cite{ICD}$$N\le\frac{C}
{12\,\pi}\int\limits_0^\infty{\rm d}r\,r^2\left[|V(r)|
\left(|V(r)|+4\,m\right)\right]^{3/2}\ ,$$ with the constant $C$
given by $C=14.107590867$ for $m>0$ or $C=6.074898097$ for
$m=0$~\cite{WL:K}.\item[3.~Upper bounds on the eigenvalues of
$\widetilde H$]arise as rather trivial consequence of the
so-called minimum--maximum theorem \cite{RS}: for every
self-adjoint operator $H$ bounded from below with (ordered)
eigenvalues $E_0\le E_1\le E_2\le\cdots$, the $d$ (ordered)
eigenvalues $\widehat E_0\le\widehat E_1\le\cdots\le\widehat
E_{d-1}$ of the operator $\widehat H$ defined by a restriction of
$H$ to a $d$-dimensional subspace of the domain of $H$ provide
upper bounds to the first $d$ eigenvalues of $H$ below the onset
of the \emph{essential\/}~spectrum,$$E_k\le\widehat E_k\qquad
\forall\quad k=0,1,2,\dots,d-1\ .$$Its eigenstates can be expanded
in terms of some convenient basis \cite{WL:T1,WL:T2}, e.g., in
products of generalized-Laguerre polynomials, $L_k^{(\gamma)}$,
and spherical harmonics ${\cal Y}$ for angular momentum~$\ell$:
\begin{align}\psi_k(\bm{x})\propto r^{\ell+\beta-1}\exp(-\mu\,r)
\,L_k^{(2\ell+2\beta)}(2\,\mu\,r)\,{\cal Y}(\Omega_{\bm{x}})\
,\qquad k\in{\mathbb N}_0\ ,&\label{LB}\\L_k^{(\gamma)}(x)\equiv
\sum_{t=0}^k\,\binom{k+\gamma}{k-t}\,\frac{(-x)^t}{t!}\ ,\qquad
\mu>0\ ,\qquad\beta>-\frac{1}{2}\ .&\nonumber\end{align}
\item[4.~Quality of eigenstates] \cite{WL:Q1,WL:Q2} tentatively
localized by whatever technique forms a crucial aspect that may be
quantified by the \emph{relativistic\/} generalization
\cite{WL:RVT1,WL:RVT2} of the virial theorem of nonrelativistic
quantum theory, which relates, for any of the eigenstates
$|\chi\rangle$ of the operator (\ref{H}), the expectation values
of the radial derivatives of both the kinetic terms and the
potential~$V(\bm{x})$:$$\left\langle\chi\left|
\frac{\bm{p}^2}{\sqrt{\bm{p}^2+m_1^2}}+
\frac{\bm{p}^2}{\sqrt{\bm{p}^2+m_2^2}}\right|\chi\right\rangle=
\left\langle\chi\left|\,\bm{x}\cdot\frac{\partial\,V}
{\partial\bm{x}}(\bm{x})\right|\chi\right\rangle.$$\end{description}
\pagebreak

All of this proved useful in locating the discrete spectra of
problems defined by spherically symmetric (central) potentials
$V(\bm{x})=V(r)$ (where $r\equiv|\bm{x}|$) of Woods--Saxon
\cite{WL:WS,WL:Q@W}, Hulth\'en \cite{WL:Q@W,WL:H}, Yukawa
\cite{WL:Y}, kink-like \cite{WL:K} or generalized-Hellmann
\cite{WL:Hm,WL:R} type. As an illustration, an application of all
these tools is sketched for the totality of generalized-Hellmann
potentials.

\section{Application: Class of Potentials Generalizing Hellmann's
Proposal}Let us take the liberty to generalize the form proposed
by Hellmann \cite{H1,H2} to a superposition$$V_{\rm H}(r)\equiv
V_{\rm C}(r)+V_{\rm Y}(r)=-\frac{\kappa}{r}-\upsilon\,
\frac{\exp(-b\,r)}{r}$$of a definitely attractive Coulomb
potential $V_{\rm C}(r)$ and a Yukawa-inspired potential $V_{\rm
Y}(r)$, with Coulomb coupling $\kappa$, and Yukawa coupling
$\upsilon$ and slope parameter $b$, subject to the constraints
$$\kappa\ge0\ ,\qquad\upsilon\gtreqqless0\ ,\qquad b>0\ .$$Within
the obtained class of \emph{generalized\/} Hellmann potentials
$V_{\rm H}(r)$, the \emph{qualitative\/} behaviour of any of its
members depends on the relative sign and size of the two coupling
strengths $\kappa$ and $\upsilon$ that govern the Coulomb and
Yukawa contributions to such generalized Hellmann~potential. These
potentials show a significant diversity: they may be unbounded
[Fig.~\ref{Pos}(a,b)] or bounded [Fig.~\ref{Pos}(c,d)] from below,
and be singular [Fig.~\ref{Pos}(a,b,d)] or finite
[Fig.~\ref{Pos}(c)] at the origin, $r=0$.

Needless to stress, using the tools sketched in Sect.~\ref{S}, it
is a breeze to rough out the~spectra of the entirety of
relativistic Hellmann problems (for some relevant details, consult
Ref.~\cite{WL:Hm}).\begin{itemize}\item Evidently, boundedness
from below is the question to be answered first and foremost:
Since \emph{any\/} of our generalized Hellmann potentials is
trivially bounded from below by conveniently selected Coulomb
potentials, suboptimal bounds on the spectrum $\sigma(\widetilde
H)$ necessarily exist for $$\kappa+\upsilon\le\frac{4}{\pi}\
.$$For the sum of couplings non-positive, i.e., $\kappa+\upsilon
\le0$, this lower bound of Coulomb origin~is easily improvable:
$V_{\rm H}(r)$ is bounded from below and hence $\sigma(\widetilde
H)$ by the minimum of $V_{\rm H}(r)$.\end{itemize}

\begin{table}[hb]\centering\caption{Binding energies $E_k-2\,m\;[m]$
($k\in{\mathbb N}_0$) of the spinless relativistic Hellmann
problem \cite{WL:Hm,WL:R}, for lowest-lying states of radial
excitation $n_{\rm r}$ and orbital angular momentum $\ell$,
assuming $b=\mu=m$ and $\beta=1$ for the potential-slope and
Laguerre-basis parameters, and trial-space dimensions around
$d\gtrapprox29$.}\label{B}\begin{tabular}{cclll}\toprule&&
\multicolumn{3}{c}{Upper bounds}\\\cline{3-5}\\[-2ex]
\multicolumn{2}{c}{Bound state}&\multicolumn{1}{c}
{$\kappa=\upsilon=\frac{1}{2}$}&\multicolumn{1}{c}{$\kappa=1$,
$\upsilon=-1$}&\multicolumn{1}{c}{$\kappa=1$, $\upsilon=-2$}\\
\cline{1-2}\\[-2ex]$n_{\rm r}$&$\ell$&\multicolumn{1}{c}{[case
Fig.~\ref{Pos}(b)]}&\multicolumn{1}{c}{[case Fig.~\ref{Pos}(c)]}&
\multicolumn{1}{c}{[case Fig.~\ref{Pos}(d)]}\\\midrule
0&0&$\quad\!\!-0.11673$&$\quad-0.17951$&$\quad-0.14410$\\
0&1&$\quad\!\!-0.01579$&$\quad-0.06294$&$\quad-0.06157$\\
0&2&$\quad\!\!-0.00616$&$\quad-0.02813$&$\quad-0.02812$\\[.8ex]
1&0&$\quad\!\!-0.02107$&$\quad-0.05464$&$\quad-0.04786$\\
1&1&$\quad\!\!-0.00509$&$\quad-0.02810$&$\quad-0.02762$\\[.8ex]
2&0&$\quad\!\!-0.00688$&$\quad-0.02566$&$\quad-0.02338$\\
\midrule\multicolumn{2}{l}{Lower bound}&$\quad\!\!-0.58578\dots$&
$\quad-1$&$\quad-0.37336\dots$\\\bottomrule\end{tabular}\end{table}

\begin{figure}[ht]\centering{\begin{tabular}{cc}
\includegraphics[scale=1.7076,clip]{ygk.eps}\hspace{1ex}&\hspace{1ex}
\includegraphics[scale=1.7076,clip]{yek.eps}\\[2ex](a)&(b)\\[4.3849ex]
\includegraphics[scale=1.7076,clip]{ye-k+.eps}\hspace{1ex}&\hspace{1ex}
\includegraphics[scale=1.7076,clip]{yl-k+.eps}\\[2ex](c)&(d)\end{tabular}}
\\[1.52405ex]\caption{Representatives of four (out of the in total
seven) different categories of generalized Hellmann potentials
$V_{\rm H}(r)$, specified by the (in)equalities of their couplings
(a) $\upsilon>\kappa$, (b) $\upsilon=\kappa$, (c)
$\upsilon=-\kappa$, and~(d) $\upsilon<-\kappa$, obtained as sums
(solid black) of Coulombic (dashed magenta) and Yukawa (dotted
blue) terms. The systematic classification of all
\emph{generalized\/} Hellmann potentials can be found in Table~1
of Ref.~\cite{WL:R}.}\label{Pos}\end{figure}\clearpage

\begin{itemize}\item The number of discrete eigenstates of
\emph{any\/} generalized-Hellmann Hamiltonian may never~be bounded
from above because any generalized Hellmann potential becomes
Coulomb-like at large distances, in the limit $r\to\infty$. More
technically, any generalized relativistic Hellmann problem fails
\cite{WL:Hm} to satisfy the prerequisite (\ref{L}) that guarantees
finiteness \cite{ICD} of the number.\item Upper bounds on the
eigenvalues corresponding to lowest-lying discrete eigenstates may
be conveniently calculated numerically upon spanning that
$d$-dimensional subspace referred to by the minimum--maximum
theorem by a Laguerre basis (\ref{LB}), as exemplified in
Table~\ref{B};~they can be optimized by increasing the dimension
$d$ and adapting the two parameters $\mu$
and/or~$\beta$.\end{itemize}

