# Lower bounds for self-adjoint Sturm–Liouville operators

Jussi Behrndt  Technische Universität Graz
Institut für Angewandte Mathematik
Steyrergasse 30
8010 Graz, Austria<EMAIL_ADDRESS>https://www.math.tugraz.at/~behrndt/ ,
Fritz Gesztesy  Department of Mathematics, Baylor University, Sid Richardson
Bldg., 1410 S. 4th Street, Waco, TX 76706, USA<EMAIL_ADDRESS>http://www.baylor.edu/math/index.php?id=935340 ,  Philipp Schmitz  Department
of Mathematics
Technische Universität Ilmenau
Postfach 100565
98648 Ilmenau
Germany<EMAIL_ADDRESS>https://www.tu-
ilmenau.de/obc/team/philipp-schmitz and  Carsten Trunk  Department of
Mathematics
Technische Universität Ilmenau
Postfach 100565
98648 Ilmenau
Germany<EMAIL_ADDRESS>https://www.tu-
ilmenau.de/funktionalanalysis

###### Abstract.

In this note we provide estimates for the lower bound of the self-adjoint
operator associated with the three-coefficient Sturm–Liouville differential
expression

$\frac{1}{r}\left(-\frac{\mathrm{d}}{\mathrm{d}x}p\frac{\mathrm{d}}{\mathrm{d}x}+q\right)$

in the weighted $L^{2}$-Hilbert space $L^{2}(\mathbb{R};rdx)$.

##  1\. Introduction

One-dimensional Schrödinger operators of the form

$H=-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+q$ (1.1)

a with real-valued potential $q$ have been studied in the mathematical and
physical literature intensively in the last century due to their particular
importance in quantum mechanics. Typically one is interested in a suitable
self-adjoint realization in $L^{2}(\mathbb{R})$ and its spectral properties,
among them estimates for lower bounds, numbers of negative eigenvalues, and
Lieb–Thirring inequalities are particularly important, see, for instance, the
recent survey [9].

The main objective of this note is to derive estimates on the lower bound of
more general Sturm-Liouville operators of the type

$T=\frac{1}{r}\left(-\frac{\mathrm{d}}{\mathrm{d}x}p\frac{\mathrm{d}}{\mathrm{d}x}+q\right)$
(1.2)

with real-valued coefficients under the standard assumptions $r,1/p,q\in
L^{1}_{\rm loc}(\mathbb{R})$ and $r,p$ positive almost everywhere. We refer
the reader to the textbooks [6], [11], [13], [15], [18], [21], [22], and [23]
for an overview and detailed study of Sturm-Liouville (resp., Schrödinger)
operators. The natural Hilbert space in this context is the weighted
$L^{2}$-space $L^{2}_{r}(\mathbb{R}):=L^{2}(\mathbb{R};rdx)$ and under some
mild additional assumptions on the coefficients one concludes that $T$ is a
semibounded self-adjoint operator in $L^{2}_{r}(\mathbb{R})$. As mentioned
above, lower bounds for the spectrum of $T$ are known for the special case
$r=p=1$, that is, $T=H$, and for completeness we provide a straightforward
estimate as a warm up in Section 2.

In the general setting it seems that a systematic study is missing and it is
the aim of this note to initiate and contribute to this circle of problems. It
is clear that the coefficients $r$ and $p$ have an essential influence on the
lower bound. If, for instance, the weight function $r=r_{0}$ is constant and
$p=1$ then formally $T=(1/r_{0})H$ and the lower bound $\min\sigma(T)$ of $T$
is simply given by $(1/r_{0})\min\sigma(H)$. This already indicates that for a
nonconstant weight function $r$ the $L^{\infty}$-norm of $1/r$ will appear in
the lower bounds, and the situation becomes much more difficult if $1/r\not\in
L^{\infty}(\mathbb{R})$, in which case we require the existence of a function
$g$ that neutralizes the behaviour of the weight function $r$ on subsets of
$\mathbb{R}$ where $r$ is small. Furthermore, the norm of the coefficient $p$
will enter in lower bound estimates and very roughly speaking $1/p$ has to be
considered in conjunction with the potential $q$. The methods and proofs in
this paper are strongly inspired by [2], where bounds on nonreal eigenvalues
of indefinite Sturm-Liouville operators are obtained.

### Acknowledgements.

Jussi Behrndt gratefully acknowledges financial support by the Austrian
Science Fund (FWF): P 33568-N. The authors wish to thank the Erwin Schrödinger
International Institute for Mathematics and Physics (ESI) at University of
Vienna, where this article was finalized during the workshop Spectral Theory
of Differential Operators in November 2022. This publication is also based
upon work from COST Action CA 18232 MAT-DYN-NET, supported by COST (European
Cooperation in Science and Technology), www.cost.eu.

##  2\. One-dimensional Schrödinger operators

As a warm up we discuss in this short section the special case $p=r=1$ and
$q\in L^{s}(\mathbb{R})$ real-valued a.e., $s\in[1,\infty]$, and derive a
lower bound for the self-adjoint Schrödinger operator $H$ in (1.1) using the
argument presented in [20, (3.5.30), p. 155–156].

We start by recalling that $q\in L^{s}(\mathbb{R})$, $s\in[1,\infty)$, implies
that $q$ is relatively form compact with respect to the free Hamiltonian
$H_{0}$ in $L^{2}(\mathbb{R})$, where

$H_{0}f=-f^{\prime\prime},\quad f\in D(H_{0})=H^{2}(\mathbb{R}),$ (2.1)

with $H^{\ell}(\mathbb{R})$, $\ell\in[0,\infty)$, the standard scale of
Sobolev spaces. This follows from the stronger statement that
$|q|^{1/2}(H_{0}+I)^{-1/2}$ satisfies (see, e.g., [16, Theorem XI.20])

$|q|^{1/2}(H_{0}+I)^{-1/2}\in\mathcal{B}_{2s}\bigl{(}L^{2}(\mathbb{R})\bigr{)}\,\text{
if }\,q\in L^{s}(\mathbb{R}),\;s\in[1,\infty),$ (2.2)

where $\mathcal{B}_{t}(\mathcal{H})$ represent the
$\ell^{t}(\mathbb{N})$-based trace ideals of compact operators in the complex,
separable Hilbert space $\mathcal{H}$. In particular,

$|q|^{1/2}(H_{0}+I)^{-1/2}\,\text{ is compact,}$ (2.3)

and hence the form sum $H$ of $H_{0}$ and $q$ is self-adjoint in
$L^{2}(\mathbb{R})$ and bounded from below. By a result of Hartman [12] and
Rellich [17] (see also [11, Theorem 8.5.2]), the boundedness from below of the
minimal operator associated with the differential expression
$-(d^{2}/dx^{2})+q$ implies that the latter is in the limit point case at
$\pm\infty$ and hence the maximal operator associated with $-(d^{2}/dx^{2})+q$
is self-adjoint in $L^{2}(\mathbb{R})$, and thus necessarily coincides with
$H$. It is clear that for $s=\infty$ the same is true as $q\in
L^{\infty}(\mathbb{R})$ is a bounded perturbation of $H_{0}$. Consequently,
$H$ is given by

$\begin{split}&Hf=-f^{\prime\prime}+qf,\\\ &f\in D(H)=\bigl{\\{}g\in
L^{2}(\mathbb{R})\,\big{|}\,g,g^{\prime}\in AC_{\rm
loc}(\mathbb{R});\,(-g^{\prime\prime}+qg)\in
L^{2}(\mathbb{R})\bigr{\\}}.\end{split}$

Property (2.3) then implies

$\sigma_{\rm ess}(H)=\sigma_{\rm ess}(H_{0})=[0,\infty),$ (2.4)

and hence it suffices to consider negative eigenvalues, which turn out to be
simple as $-(d^{2}/dx^{2})+q$ is in the limit point case at $\pm\infty$. We
consider an eigenvalue $\lambda<0$ of $H$ and denote the corresponding
eigenfunction by $f_{\lambda}$. From
$-f_{\lambda}^{\prime\prime}+qf_{\lambda}=\lambda f_{\lambda}$ one concludes

$f_{\lambda}=-\left(-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-\lambda\right)^{-1}qf_{\lambda}$

and using the corresponding Green’s function we obtain

$\|f_{\lambda}\|_{2}=\frac{1}{2\sqrt{-\lambda}}\bigl{\|}e^{-\sqrt{-\lambda}|\,\cdot\,|}\ast
qf_{\lambda}\bigr{\|}_{2}\leq\frac{1}{2\sqrt{-\lambda}}\bigl{\|}e^{-\sqrt{-\lambda}|\,\cdot\,|}\bigr{\|}_{t}\|qf_{\lambda}\|_{t^{\prime}},$
(2.5)

where Young’s inequality,111Explicitly, $\|f\ast
g\|_{\alpha}\leq\|f\|_{\beta}\|g\|_{\gamma}$,
$1\leq\alpha,\beta,\gamma\leq\infty$, $1+\alpha^{-1}=\beta^{-1}+\gamma^{-1}$.
with $1/t+1/t^{\prime}=1+1/2$ was applied in the last step. Hölder’s
inequality then yields
$\|qf_{\lambda}\|_{t^{\prime}}\leq\|q\|_{s}\|f_{\lambda}\|_{2}$ for
$1/t^{\prime}=1/s+1/2$ and hence,

$\sqrt{-\lambda}\leq\frac{1}{2}\bigl{\|}e^{-\sqrt{-\lambda}|\,\cdot\,|}\bigr{\|}_{t}\|q\|_{s}=\frac{1}{2}\left(\frac{2}{t\sqrt{-\lambda}}\right)^{\frac{1}{t}}\|q\|_{s}$
(2.6)

if $t\in(1,\infty)$, that is, $s\in(1,\infty)$. As $1/t=1-1/s$ it follows for
$s\in(1,\infty)$ that

$(-\lambda)^{\frac{2s-1}{2s}}\leq
2^{-\frac{1}{s}}\left(\frac{s-1}{s}\right)^{\frac{s-1}{s}}\|q\|_{s}$

and hence

$\min\sigma(H)\geq-2^{-\frac{2}{2s-1}}\left(\frac{s-1}{s}\right)^{\frac{2(s-1)}{2s-1}}\|q\|_{s}^{\frac{2s}{2s-1}}\,{\text{
for }\,s\in(1,\infty)}.$ (2.7)

It is easy to see that the lower bound (2.7) also remains valid for $s=1$
(indeed, inequality (2.5) and the first inequality in (2.6) apply with $s=1$,
$t=\infty$, $1/t^{\prime}=3/2$) in which case one obtains

$\min\sigma(H)\geq-(1/4)\|q\|_{1}^{2},$ (2.8)

and obviously applies to $s=\infty$, implying

$\min\sigma(H)\geq-\|q\|_{\infty}.$ (2.9)

We mention that the bound (2.7) and (2.8) coincide with [5, Corollary 14.3.11
and Corollary 14.3.12] and that the above argument also leads to bounds for
Schrödinger operators with complex potentials $q\in L^{s}(\mathbb{R})$,
$s\in[1,\infty)$ (see, in particular, [1] for the case $s=1$). In the context
of Schrödinger operators with complex-valued potentials we also refer, for
instance, to [4], [7], [8], and [10].

######  Remark 2.1.

We mention that Lieb–Thirring inequalities (see, e.g., [9], [14] and the
extensive literature cited therein) also lead to lower bounds of $H$. More
specifically, for $s=3/2$ one can compare with the one-particle constant
$L_{1}^{(1)}=4\big{/}\big{[}3^{3/2}\pi\big{]}=0.24503$ in [9, Section 3.2]:
The corresponding constant in (2.7) equals $2^{-1}3^{-1/2}=0.28867$.
Historically, we note that Barnes, Brascamp, and Lieb [3] derived a lower
bound for the ground state energy of (multi-dimensional) Schrödinger operators
already in 1976.

##  3\. Main results

Now we consider the general Sturm-Liouville differential expression

$\tau=\frac{1}{r}\left(-\frac{\mathrm{d}}{\mathrm{d}x}p\frac{\mathrm{d}}{\mathrm{d}x}+q\right)$
(3.1)

on $\mathbb{R}$ with real-valued coefficients under the standard assumptions,

$r,1/p,q\in L^{1}_{\rm loc}(\mathbb{R}),$ (3.2)

and we assume from now on that Hypothesis 3.1 below is satisfied. In the
following $L^{1}_{\mathrm{u}}(\mathbb{R})$ denotes the normed space of
uniformly locally integrable functions, that is,

$L^{1}_{\mathrm{u}}(\mathbb{R})=\big{\\{}h\in
L^{1}_{\mathrm{loc}}(\mathbb{R}):\lVert
h\rVert_{\mathrm{u}}<\infty\big{\\}},\qquad\lVert
h\rVert_{\mathrm{u}}=\sup_{n\in\mathbb{Z}}\int_{n}^{n+1}\lvert
h(t)\rvert\,\mathrm{d}t.$

######  Hypothesis 3.1.

The real coefficients $p$, $q$ and $r$ of $\tau$ satisfy the following:

  1. (a)

$p(x)>0$ for a. a. $x\in\mathbb{R}$ and $1/p\in L^{\eta}(\mathbb{R})$ for some
$\eta\in[1,\infty]$;

  2. (b)

$q\in L^{1}_{\mathrm{u}}(\mathbb{R})$;

  3. (c)

$r(x)>0$ for a. a. $x\in\mathbb{R}$ and there exist $a$, $b\in\mathbb{R}$ with
$a<b$ such that

$\operatorname{ess\,inf}_{t\in\mathbb{R}\setminus[a,b]}r(t)>0.$ (3.3)

It is known that the differential expression $\tau$ is in the limit-point case
at both singular endpoints $\pm\infty$, and the corresponding maximal operator

$\begin{split}&Tf=\tau
f=\frac{1}{r}\left(-(pf^{\prime})^{\prime}+qf\right),\\\ &f\in
D(T)=\bigl{\\{}g\in L^{2}_{r}(\mathbb{R})\,\big{|}\,g,pg^{\prime}\in AC_{\rm
loc}(\mathbb{R});\,\tau g\in L^{2}_{r}(\mathbb{R})\bigr{\\}},\end{split}$

is self-adjoint in the weighted $L^{2}$-space $L^{2}_{r}(\mathbb{R})$ and
semibounded from below; cf. [2, Lemma A.2]. Our main goal is to derive
estimates for the lower bound $\min\sigma(T)$ of $T$. For a nonnegative
function $g\in L^{\infty}(\mathbb{R})$ the set

$\Omega_{g}:=\\{x\in\mathbb{R}\,|\,r(x)g(x)<1\\}$

and its Lebesgue measure $\mu(\Omega_{g})$ will appear in the lower bound
estimates in our main results below. In the particular case $1/r\in
L^{\infty}(\mathbb{R})$ one can choose $g=1/r$ and hence $\Omega_{g}$ becomes
a Lebesgue null set, which leads to more explicit lower bound estimates. We
also mention that for any $\varepsilon>0$ there exists a (constant)
nonnegative function $g_{\varepsilon}\in L^{\infty}(\mathbb{R})$ such that
$\mu(\Omega_{g_{\varepsilon}})<\varepsilon$; this follows from

$\begin{split}\lim_{n\rightarrow\infty}\mu(\\{x\in\mathbb{R}\,|\,r(x)<1/n\\})&=\mu\left(\bigcap_{n=1}^{\infty}\\{x\in\mathbb{R}\,|\,r(x)<1/n\\}\right)\\\
&=\mu(\\{x\in\mathbb{R}\,|\,r(x)=0\\})=0.\end{split}$

The first main result requires only minimal assumptions on the potential in
Hypothesis 3.1, that is, $q\in L^{1}_{\mathrm{u}}(\mathbb{R})$, but we have to
assume that $1/p\in L^{\infty}(\mathbb{R})$. We shall denote the negative part
of the potential $q$ by $q_{-}$.

######  Theorem 3.2.

In addition to Hypothesis 3.1, assume that $1/p\in L^{\infty}(\mathbb{R})$,
let

$\alpha=2\lVert q_{-}\|_{\mathrm{u}}+4\lVert 1/p\rVert_{\infty}\lVert
q_{-}\|_{\mathrm{u}}^{2}\quad\text{and}\quad\beta=\left(4\lVert
1/p\rVert_{\infty}\alpha\right)^{1/2},$ (3.4)

and choose a nonnegative function $g\in L^{\infty}(\mathbb{R})$ such that
$\mu(\Omega_{g})\beta<1$. Then

$\min\sigma(T)\geq\frac{-\alpha\lVert
g\rVert_{\infty}}{1-\mu(\Omega_{g})\beta}$

and in the special case $1/r\in L^{\infty}(\mathbb{R})$ the choice $g=1/r$
implies $\mu(\Omega_{g})=0$ and

$\min\sigma(T)\geq-\bigl{(}2\lVert q_{-}\|_{\mathrm{u}}+4\lVert
1/p\rVert_{\infty}\lVert q_{-}\|_{\mathrm{u}}^{2}\bigr{)}\lVert
1/r\rVert_{\infty}.$

In the next result we consider the case $q_{-}\in L^{s}(\mathbb{R})$,
$s\in[1,\infty]$, and $1/p\in L^{\infty}(\mathbb{R})$.

######  Theorem 3.3.

In addition to Hypothesis 3.1, assume that $1/p\in L^{\infty}(\mathbb{R})$ and
$q_{-}\in L^{s}(\mathbb{R})$ for some $s\in[1,\infty]$, let

$\alpha=\begin{cases}\|q_{-}\|_{s}\beta^{\frac{1}{s}}&\text{if}\,\,s\in[1,\infty),\\\\[4.30554pt]
\|q_{-}\|_{\infty}&\text{if}\,\,s=\infty,\end{cases}\quad\text{and}\quad\beta=\begin{cases}\left(4\lVert
1/p\rVert_{\infty}\lVert
q_{-}\|_{s}\right)^{\frac{s}{2s-1}}&\text{if}\,\,s\in[1,\infty),\\\\[4.30554pt]
(4\lVert 1/p\rVert_{\infty}\lVert
q_{-}\|_{\infty})^{1/2}&\text{if}\,\,s=\infty,\end{cases}$ (3.5)

and choose a nonnegative function $g\in L^{\infty}(\mathbb{R})$ such that
$\mu(\Omega_{g})\beta<1$. Then

$\min\sigma(T)\geq\frac{-\alpha\lVert
g\rVert_{\infty}}{1-\mu(\Omega_{g})\beta}$

and in the special case $1/r\in L^{\infty}(\mathbb{R})$ the choice $g=1/r$
implies $\mu(\Omega_{g})=0$ and

$\min\sigma(T)\geq-\bigl{(}4\lVert
1/p\rVert_{\infty}\bigr{)}^{\frac{1}{2s-1}}\lVert
q_{-}\|_{s}^{\frac{2s}{2s-1}}\lVert 1/r\rVert_{\infty}\quad\text{if
}s\in[1,\infty).$ (3.6)

If $s=\infty$ we have

$\min\sigma(T)\geq-\|q_{-}\|_{\infty}\lVert 1/r\rVert_{\infty}.$

######  Remark 3.4.

The bounds in Theorem 3.3 above are not optimal. In fact, in the special case
$p=r=1$ and $q_{-}\in L^{s}(\mathbb{R})$ for some $s\in[1,\infty)$ the bound
in (3.6) becomes

$\min\sigma(T)\geq\begin{cases}-4\lVert
q_{-}\|_{1}^{2}&\text{if}\,\,s=1,\\\\[4.30554pt] -2^{\frac{2}{2s-1}}\lVert
q_{-}\|_{s}^{\frac{2s}{2s-1}}&\text{if}\,\,s\in(1,\infty),\end{cases}$

while (2.7) (or [5, Corollary 14.3.11 and Corollary 14.3.12]) show that

$\min\sigma(T)\geq\begin{cases}-\lVert
q\rVert_{1}^{2}\big{/}4&\text{if}\,\,s=1,\\\\[4.30554pt]
-2^{-\frac{2}{2s-1}}\left(\frac{s-1}{s}\right)^{\frac{2(s-1)}{2s-1}}\lVert
q\rVert_{s}^{\frac{2s}{2s-1}}&\text{if}\,\,s\in(1,\infty).\end{cases}$ (3.7)

$\diamond$

In the following theorem we deal with $q_{-}\in L^{s}(\mathbb{R})$,
$s\in[1,\infty]$, and $1/p\in L^{\eta}(\mathbb{R})$, $\eta\in[1,\infty)$.

######  Theorem 3.5.

In addition to Hypothesis 3.1, assume that $1/p\in L^{\eta}(\mathbb{R})$ for
some $\eta\in[1,\infty)$ and $q_{-}\in L^{s}(\mathbb{R})$ for some
$s\in[1,\infty]$ such that $\eta+s>2$ if $s\not=\infty$. Let $\alpha$ be as in
(3.5) and

$\beta=\begin{cases}\Bigl{(}\Bigl{(}\frac{2\eta-1}{\eta}\Bigr{)}^{2}\lVert
1/p\rVert_{\eta}\lVert q_{-}\|_{s}\Bigr{)}^{\frac{\eta s}{2\eta
s-\eta-s}}&\text{if}\,\,s\in[1,\infty),\\\\[4.30554pt]
\Bigl{(}\Bigl{(}\frac{2\eta-1}{\eta}\Bigr{)}^{2}\lVert 1/p\rVert_{\eta}\lVert
q_{-}\|_{\infty}\Bigr{)}^{\frac{\eta}{2\eta-1}}&\text{if}\,\,s=\infty,\end{cases}$
(3.8)

and choose a nonnegative function $g\in L^{\infty}(\mathbb{R})$ such that
$\mu(\Omega_{g})\beta<1$. Then

$\min\sigma(T)\geq\frac{-\alpha\lVert
g\rVert_{\infty}}{1-\mu(\Omega_{g})\beta}$

and in the special case $1/r\in L^{\infty}(\mathbb{R})$ the choice $g=1/r$
implies $\mu(\Omega_{g})=0$ and

$\min\sigma(T)\geq-\Bigl{(}\Bigl{(}\frac{2\eta-1}{\eta}\Bigr{)}^{2}\lVert
1/p\rVert_{\eta}\Bigr{)}^{\frac{\eta}{2\eta s-\eta-s}}\lVert
q_{-}\|_{s}^{\frac{2\eta s-s}{2\eta s-\eta-s}}\lVert
1/r\rVert_{\infty}\quad\text{if }s\in[1,\infty).$

If $s=\infty$ we have

$\min\sigma(T)\geq-\|q_{-}\|_{\infty}\lVert 1/r\rVert_{\infty}.$

The following proposition provides a simple condition for nonnegativity of
$T$.

######  Proposition 3.6.

Suppose that Hypothesis 3.1 holds and assume that $1/p,q_{-}\in
L^{1}(\mathbb{R})$. If $\lVert 1/p\rVert_{1}\lVert q_{-}\|_{1}<1$ then
$\min\sigma(T)\geq 0$.

######  Remark 3.7.

If the coefficients $p$, $q$ and $r$ in Hypothesis 3.1 are restricted to the
half line $(0,\infty)$ and $T_{+}$ denotes the self-adjoint realization of the
(restricted) differential expression $\tau$ in $L^{2}((0,\infty))$ with
Dirichlet boundary conditions at the regular endpoint $0$, then the lower
bound estimates above remain valid for $T_{+}$. In fact, the Dirichlet
boundary condition at $0$ ensures that the boundary term in the integration by
parts formula for $f\in D(T_{+})$ vanishes and hence the proofs in the next
section (see, e.g. (4.4)) extend directly to the half line case.

##  4\. Proofs

In this section we prove our main results. It will always be assumed that the
coefficients $p,q,r$ satisfy Hypothesis 3.1. The first three items of the
following useful statement can be found, for instance, in [2, Lemma A.2]. The
last item follows from [2, Lemma A.1] and the first item.

######  Lemma 4.1.

Assume Hypothesis 3.1, then the following assertions hold for all $f$, $g\in
D(T)$:

  * (i)

$f$, $\sqrt{p}f^{\prime}\in L^{2}(\mathbb{R})$ and $qf^{2}\in
L^{1}(\mathbb{R})$;

  * (ii)

there exists a sequence $(x_{n})_{n\in\mathbb{Z}}$ in $\mathbb{R}$ satisfying
$\lim_{n\rightarrow\infty}x_{n}=\infty$ and
$\lim_{n\rightarrow-\infty}x_{n}=-\infty$ such that $\lim_{\lvert
n\rvert\rightarrow\infty}f(x_{n})=0$;

  * (iii)

$\lim_{\lvert x\rvert\rightarrow\infty}(pf^{\prime})(x)\overline{g(x)}=0$.

  * (iv)

$f\in L^{\infty}(\mathbb{R})$.

For our estimates of the lower bound of $T$ it is convenient to reduce the
considerations to the set

$D_{-}(T):=\\{f\in D(T)\,|\,(Tf,f)_{r}\leq 0\\},$ (4.1)

where $(\cdot,\cdot)_{r}$ stands for the weighted inner product corresponding
to $L_{r}^{2}(\mathbb{R})$. The potential $q$ is decomposed in its positive
part $q_{+}$ and negative part $q_{-}$, i. e.

$q=q_{+}-q_{-},\quad\text{where}\quad q_{+}:=\frac{\lvert
q\rvert+q}{2}\quad\text{and}\quad q_{-}:=\frac{\lvert q\rvert-q}{2}.$ (4.2)

######  Lemma 4.2.

Assuming Hypothesis 3.1, every function $f\in D_{-}(T)$ satisfies

$\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}\leq\lVert
q_{-}f^{2}\rVert_{1}\quad\text{and}\quad\lVert qf^{2}\rVert_{1}\leq 2\lVert
q_{-}f^{2}\rVert_{1}.$ (4.3)

Moreover, the inequality $\lVert q_{-}f^{2}\lVert_{1}\leq\lVert
q_{+}f^{2}\|_{1}$ implies $\lVert\sqrt{p}f^{\prime}\rVert_{2}=0$.

###### Proof.

For $f\in D_{-}(T)$ integration by parts together with Lemma 4.1 (i) and (iii)
yields

$\begin{split}0\geq(Tf,f)_{r}&=\int_{\mathbb{R}}p(t)\lvert
f^{\prime}(t)\rvert^{2}\,\mathrm{d}t+\int_{\mathbb{R}}q(t)\lvert
f(t)\rvert^{2}\,\mathrm{d}t\\\ &=\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}+\lVert
q_{+}f^{2}\rVert_{1}-\lVert q_{-}f^{2}\rVert_{1}.\end{split}$ (4.4)

This implies $\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}\leq\lVert
q_{-}f^{2}\rVert_{1}$ and $\lVert q_{+}f^{2}\rVert_{1}\leq\lVert
q_{-}f^{2}\rVert_{1}$. Therefore, with $\lvert q\rvert=q_{+}+q_{-}$ we have

$\lVert qf^{2}\rVert_{1}=\lVert q_{+}f^{2}\rVert_{1}+\lVert
q_{-}f^{2}\rVert_{1}\leq 2\lVert q_{-}f^{2}\rVert_{1}.$

If $\lVert q_{-}f^{2}\rVert_{1}\leq\lVert q_{+}f^{2}\rVert_{1}$ holds, then
(4.4) implies $\lVert\sqrt{p}f^{\prime}\rVert_{2}=0$. ∎

######  Lemma 4.3.

In addition to Hypothesis 3.1, assume that there are constants $\alpha\geq 0$,
$\beta\geq 0$ and a nonnegative function $g\in L^{\infty}(\mathbb{R})$ such
that

  1. (i)

$\lVert q_{-}f^{2}\rVert_{1}\leq\alpha\|f\rVert_{2}^{2}$ and
$\|f\rVert_{\infty}^{2}\leq\beta\|f\rVert_{2}^{2}$ for all $f\in D_{-}(T)$;

  2. (ii)

$\mu(\Omega_{g})\beta<1$.

Then one has

$\min\sigma(T)\geq\frac{-\alpha\lVert
g\rVert_{\infty}}{1-\mu(\Omega_{g})\beta}.$ (4.5)

###### Proof.

Let $f\in D_{-}(T)$. Then one has

$\begin{split}\lVert g\rVert_{\infty}(f,f)_{r}&=\lVert
g\rVert_{\infty}\int_{\mathbb{R}}\lvert
f(t)\rvert^{2}r(t)\,\mathrm{d}t\geq\int_{\mathbb{R}}\lvert
f(t)\rvert^{2}r(t)g(t)\,\mathrm{d}t\\\\[6.0pt]
&\geq\int_{\mathbb{R}\setminus\Omega_{g}}\lvert
f(t)\rvert^{2}r(t)g(t)\,\mathrm{d}t\geq\|f\rVert_{2}^{2}-\int_{\Omega_{g}}\lvert
f(t)\rvert^{2}\,\mathrm{d}t\\\\[6.0pt]
&\geq\|f\rVert_{2}^{2}-\mu(\Omega_{g})\|f\rVert_{\infty}^{2}\geq\bigl{(}1-\mu(\Omega_{g})\beta\bigr{)}\|f\rVert_{2}^{2}.\end{split}$
(4.6)

Further, we have by (4.4)

$(Tf,f)_{r}=\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}+\lVert
q_{+}f^{2}\rVert_{1}-\lVert q_{-}f^{2}\rVert_{1}\geq-\lVert
q_{-}f^{2}\rVert_{1}\geq-\alpha\|f\rVert_{2}^{2}.$

This together with (4.6) yields

$(Tf,f)_{r}\geq-\frac{\alpha\lVert
g\rVert_{\infty}}{1-\mu(\Omega_{g})\beta}(f,f)_{r}.$ (4.7)

Obviously, the inequality in (4.7) holds also for $f\in D(T)\setminus
D_{-}(T)$ and, thus, for all $f\in D(T)$. This implies (4.5) ∎

Next we recall estimates on the $L^{\infty}$-norm of functions in $D(T)$ from
[2].

######  Lemma 4.4.

Assume Hypothesis 3.1. Then the following assertions hold for all $f\in D(T)$.

  1. (i)

If $1/p\in L^{\eta}(\mathbb{R})$, where $\eta\in[1,\infty)$, then

$\|f\rVert_{\infty}\leq\left(\frac{2\eta-1}{\eta}\sqrt{\lVert
1/p\rVert_{\eta}}\lVert\sqrt{p}f^{\prime}\rVert_{2}\right)^{\frac{\eta}{2\eta-1}}\|f\rVert_{2}^{\frac{\eta-1}{2\eta-1}}.$
(4.8)

  2. (ii)

If $1/p\in L^{\infty}(\mathbb{R})$ then

$\|f\rVert_{\infty}\leq\left(2\sqrt{\lVert
1/p\rVert_{\infty}}\lVert\sqrt{p}f^{\prime}\rVert_{2}\|f\rVert_{2}\right)^{1/2}.$
(4.9)

Moreover, for every $\varepsilon>0$ and all $n\in\mathbb{Z}$ one has

$\sup_{t\in[n,n+1]}\lvert f(t)\rvert^{2}\leq\varepsilon\lVert
1/p\rVert_{\infty}\int_{n}^{n+1}p(t)\lvert
f^{\prime}(t)\rvert^{2}\,\mathrm{d}t+\Bigl{(}1+\frac{1}{\varepsilon}\Bigr{)}\int_{n}^{n+1}\lvert
f(t)\rvert^{2}\,\mathrm{d}t.$ (4.10)

###### Proof.

The estimates (4.8) and (4.9) are proved in [2, Lemma 4.1]. For the
convenience of the reader we verify the estimate (4.10), which is a variant of
[19, Lemma 9.32]. Let $\varepsilon>0$ and $n\in\mathbb{Z}$. Then for $f\in
D(T)$ and $x$, $y\in[n,n+1]$

$\lvert f(x)\rvert^{2}=\lvert f(y)\rvert^{2}+2\,\text{\rm
Re}\int_{y}^{x}f^{\prime}(t)\overline{f(t)}\,\mathrm{d}t.$

By the mean value theorem we can choose $y$ such that $\lvert
f(y)\rvert^{2}=\int_{n}^{n+1}\lvert f(t)\rvert^{2}\,\mathrm{d}t$. Thus, by the
Cauchy–Schwarz inequality and $2\alpha\beta\leq\alpha^{2}+\beta^{2}$ for
$\alpha$, $\beta\in\mathbb{R}$ we obtain

$\displaystyle\lvert f(x)\rvert^{2}$ $\displaystyle\leq\int_{n}^{n+1}\lvert
f(t)\rvert^{2}\,\mathrm{d}t$
$\displaystyle\quad+2\left(\frac{1}{\varepsilon}\int_{n}^{n+1}\lvert
f(t)\rvert^{2}\,\mathrm{d}t\right)^{1/2}\cdot\left(\lVert
1/p\rVert_{\infty}\varepsilon\int_{n}^{n+1}p(t)\lvert
f^{\prime}(t)\rvert^{2}\,\mathrm{d}t\right)^{1/2}$
$\displaystyle\leq\varepsilon\lVert 1/p\rVert_{\infty}\int_{n}^{n+1}p(t)\lvert
f^{\prime}(t)\rvert^{2}\,\mathrm{d}t+\Bigl{(}1+\frac{1}{\varepsilon}\Bigr{)}\int_{n}^{n+1}\lvert
f(t)\rvert^{2}\,\mathrm{d}t,$

which leads to (4.10). ∎

For the proofs of Theorems 3.2 – 3.5 it is no restriction to consider $f\in
D_{-}(T)\setminus\\{0\\}$ and to assume that $q_{-}$ is positive on a set of
positive Lebesgue measure.

###### Proof of Theorem 3.2.

Let $1/p\in L^{\infty}(\mathbb{R})$ and consider $\alpha$, $\beta$ as in
(3.4). Choose $\varepsilon=(2\lVert q_{-}\|_{\mathrm{u}}\lVert
1/p\rVert_{\infty})^{-1}>0$. The estimate in (4.10) of Lemma 4.4 yields

$\begin{split}\lVert q_{-}f^{2}\rVert_{1}&=\int_{\mathbb{R}}q_{-}(t)\lvert
f(t)\rvert^{2}\,\mathrm{d}t\\\\[6.0pt] &\leq\lVert
q_{-}\|_{\mathrm{u}}\sum_{n\in\mathbb{Z}}\sup_{t\in[n,n+1]}\lvert
f(t)\rvert^{2}\\\\[6.0pt] &\leq\lVert
q_{-}\|_{\mathrm{u}}\left(\varepsilon\lVert
1/p\rVert_{\infty}\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}+\left(1+\frac{1}{\varepsilon}\right)\|f\rVert_{2}^{2}\right)\\\\[6.0pt]
&=\frac{1}{2}\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}+\bigl{(}\lVert
q_{-}\|_{\mathrm{u}}+2\lVert 1/p\rVert_{\infty}\lVert
q_{-}\|_{\mathrm{u}}^{2}\bigr{)}\|f\rVert_{2}^{2}\\\\[6.0pt]
&=\frac{1}{2}\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}+\frac{\alpha}{2}\|f\rVert_{2}^{2}.\end{split}$
(4.11)

Together with Lemma 4.2 we obtain

$\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}=2\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}-\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}\leq
2\lVert
q_{-}f^{2}\rVert_{1}-\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}\leq\alpha\|f\rVert_{2}^{2}.$

With (4.9) in Lemma 4.4 and (4.11) we see

$\|f\rVert_{\infty}^{2}\leq 2\sqrt{\lVert
1/p\rVert_{\infty}\alpha}\|f\rVert_{2}^{2}=\beta\|f\rVert_{2}^{2}\quad\text{and}\quad\lVert
q_{-}f^{2}\rVert_{1}\leq\alpha\|f\rVert_{2}^{2}$

and hence Lemma 4.3 leads to the statements in Theorem 3.2. ∎

###### Proof of Theorem 3.5.

Suppose that $1/p\in L^{\eta}(\mathbb{R})$ and $q_{-}\in L^{s}(\mathbb{R})$,
where $\eta$, $s\in[1,\infty)$ with $\eta+s>2$. Since $\eta+s>2$ we obtain

$2\eta s-\eta-s=\eta(s-1)+s(\eta-1)\geq s-1+\eta-1>0.$

Let $\alpha$ and $\beta$ as in (3.5) and (3.8), respectively. From Hölder’s
inequality we obtain

$\begin{split}\lVert
q_{-}f^{2}\rVert_{1}&\leq\|f\rVert_{\infty}^{\frac{2}{s}}\int_{\mathbb{R}}\lvert
q_{-}(t)\rvert\lvert f(t)\rvert^{\frac{2(s-1)}{s}}\,\mathrm{d}t\\\\[6.0pt]
&\leq\|f\rVert_{\infty}^{\frac{2}{s}}\left(\int_{\mathbb{R}}\lvert
q_{-}(t)\rvert^{s}\,\mathrm{d}t\right)^{\frac{1}{s}}\left(\int_{\mathbb{R}}\lvert
f(t)\rvert^{2}\,\mathrm{d}t\right)^{\frac{s-1}{s}}\\\\[6.0pt] &=\lVert
q_{-}\|_{s}\|f\rVert_{\infty}^{\frac{2}{s}}\|f\rVert_{2}^{\frac{2(s-1)}{s}}.\end{split}$
(4.12)

Thus, together with Lemma 4.4 (i) and Lemma 4.2 we obtain

$\displaystyle\|f\rVert_{\infty}^{2}$
$\displaystyle=\left(\frac{\|f\rVert_{\infty}^{\frac{2(2\eta-1)}{\eta}}}{\|f\rVert_{\infty}^{\frac{2}{s}}}\right)^{\frac{\eta
s}{2\eta s-\eta-s}}\leq\left(\frac{\left(\frac{2\eta-1}{\eta}\right)^{2}\lVert
1/p\rVert_{\eta}\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}\|f\rVert_{2}^{\frac{2(\eta-1)}{\eta}}}{\|f\rVert_{\infty}^{\frac{2}{s}}}\right)^{\frac{\eta
s}{2\eta s-\eta-s}}$
$\displaystyle\leq\left(\left(\frac{2\eta-1}{\eta}\right)^{2}\lVert
1/p\rVert_{\eta}\lVert q_{-}\|_{s}\right)^{\frac{\eta s}{2\eta
s-\eta-s}}\|f\rVert_{2}^{2}=\beta\|f\rVert_{2}^{2}.$

The estimate from (4.12) yields

$\lVert q_{-}f^{2}\rVert_{1}\leq\lVert
q_{-}\|_{s}\beta^{\frac{1}{s}}\|f\rVert_{2}^{2}=\alpha\|f\rVert_{2}^{2}.$

Now consider the case $q_{-}\in L^{\infty}(\mathbb{R})$. Choose $\alpha$ and
$\beta$ as in (3.5) and (3.8), respectively. Observe that

$\lVert q_{-}f^{2}\rVert_{1}\leq\lVert
q_{-}\|_{\infty}\|f\rVert_{2}^{2}=\alpha\|f\rVert_{2}^{2}.$ (4.13)

Lemma 4.4 (i) in combination with Lemma 4.2 and (4.13) leads to

$\begin{split}\|f\rVert_{\infty}^{2}&\leq\left(\left(\frac{2\eta-1}{\eta}\right)^{2}\lVert
1/p\rVert_{\eta}\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}\right)^{\frac{\eta}{2\eta-1}}\|f\rVert_{2}^{\frac{2(\eta-1)}{2\eta-1}}\\\\[6.0pt]
&\leq\left(\left(\frac{2\eta-1}{\eta}\right)^{2}\lVert 1/p\rVert_{\eta}\lVert
q_{-}\|_{\infty}\right)^{\frac{\eta}{2\eta-1}}\|f\rVert_{2}^{2}=\beta\|f\rVert_{2}^{2}.\end{split}$

Now Theorem 3.5 follows from Lemma 4.3. ∎

###### Proof of Theorem 3.3.

Consider first the case $1/p\in L^{\infty}(\mathbb{R})$ and $q_{-}\in
L^{s}(\mathbb{R})$, where $s\in[1,\infty)$. Let $\alpha$ and $\beta$ be as in
(3.5). Again Hölder’s inequality yields (4.12). Lemma 4.4 (ii), (4.12) and
Lemma 4.2 imply

$\begin{split}\|f\rVert_{\infty}^{2}&=\left(\frac{\|f\rVert_{\infty}^{4}}{\|f\rVert_{\infty}^{\frac{2}{s}}}\right)^{\frac{s}{2s-1}}\leq\left(\frac{4\lVert
1/p\rVert_{\infty}\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}\|f\rVert_{2}^{2}}{\|f\rVert_{\infty}^{\frac{2}{s}}}\right)^{\frac{s}{2s-1}}\\\\[6.0pt]
&\leq\left(4\lVert 1/p\rVert_{\infty}\lVert
q_{-}\|_{s}\right)^{\frac{s}{2s-1}}\|f\rVert_{2}^{2}=\beta\|f\rVert_{2}^{2}.\end{split}$

By applying this to the estimate in (4.12) we arrive at

$\lVert q_{-}f^{2}\rVert_{1}\leq\lVert
q_{-}\|_{s}\beta^{\frac{1}{s}}\|f\rVert_{2}^{2}=\alpha\|f\rVert_{2}^{2},$

and again the statements in Theorem 3.3 follow from Lemma 4.3.

The assertion for $1/p$, $q_{-}\in L^{\infty}(\mathbb{R})$ follows in a
similar way. Consider $\alpha$, $\beta$ in (3.5). As before (4.13) holds.
Lemma 4.4 (ii) in combination with Lemma 4.2 and (4.13) implies

$\|f\rVert_{\infty}^{2}\leq 2\sqrt{\lVert
1/p\rVert_{\infty}}\lVert\sqrt{p}f^{\prime}\rVert_{2}\|f\rVert_{2}\leq
2\sqrt{\lVert 1/p\rVert_{\infty}\lVert
q_{-}\|_{\infty}}\|f\rVert_{2}^{2}=\beta\|f\rVert_{2}^{2}.\qed$

###### Proof of Proposition 3.6.

Let $f\in D_{-}(T)$. In the case $1/p$, $q_{-}\in L^{1}(\mathbb{R})$ Lemma 4.2
and Lemma 4.4 (i) yield

$\|f\rVert_{\infty}^{2}\leq\lVert
1/p\rVert_{1}\lVert\sqrt{p}f^{\prime}\rVert_{2}^{2}\leq\lVert
1/p\rVert_{1}\lVert q_{-}f^{2}\rVert_{1}\leq\lVert 1/p\rVert_{1}\lVert
q_{-}\|_{1}\|f\rVert_{\infty}^{2}.$

If $\lVert 1/p\rVert_{1}\lVert q_{-}\|_{1}<1$, then $\|f\rVert_{\infty}=0$ and
hence $D_{-}(T)=\\{0\\}$. This implies $(Tf,f)_{r}\geq 0$ for all $f\in D(T)$
and hence $\min\sigma(T)\geq 0$. ∎

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