Indefinite Sturm–Liouville Operators in Polar Form

Abstract

We consider the indefinite Sturm–Liouville differential expression

$$\begin{aligned} {\mathfrak {a}}(f):= - \frac{1}{w}\left( \frac{1}{r} f' \right) ', \end{aligned}$$

where ${\mathfrak {a}}$ is defined on a finite or infinite open interval I with $0\in I$ and the coefficients r and w are locally summable and such that r(x) and $({\text {sgn}}\,x) w(x)$ are positive a.e. on I. With the differential expression ${\mathfrak {a}}$ we associate a nonnegative self-adjoint operator A in the Krein space $L^2_w(I)$ which is viewed as a coupling of symmetric operators in Hilbert spaces related to the intersections of I with the positive and the negative semi-axis. For the operator A we derive conditions in terms of the coefficients w and r for the existence of a Riesz basis consisting of generalized eigenfunctions of A and for the similarity of A to a self-adjoint operator in a Hilbert space $L^2_{|w|}(I)$. These results are obtained as consequences of abstract results about the regularity of critical points of nonnegative self-adjoint operators in Krein spaces which are couplings of two symmetric operators acting in Hilbert spaces.

Appendix A: Some Results from Karamata’s Theory

In Appendix we present the definitions and the results from Karamata’s theory of regularly varying functions that we use in the paper. Standard references for Karamata’s theory are [12] and [87]. For completeness we include a few standard results from Karamata’s theory and some of them are reformulated to fit our needs. In addition, we present Theorem A.6 and Corollary A.7 that seem to be new.

1.1 Definitions and basic results

First we give definitions of regularly varying functions.

Definition A.1

Let $a,\alpha \in {\mathbb {R}}$ with $a > 0$. A measurable function $f:(0,a]\rightarrow {\mathbb {R}}_+$ is called regularly varying at 0 from the right with index $\alpha$ if the following condition is satisfied:

$$\begin{aligned} \text {for all} \quad \lambda \in {\mathbb {R}}_+ \qquad \text {we have} \qquad \lim _{x\downarrow 0} \frac{f(\lambda x)}{f(x)} = \lambda ^{\alpha }. \end{aligned}$$

When $\alpha = 0$ the function f is called slowly varying at 0 from the right.

A measurable function $g:[a,+\infty ) \rightarrow {\mathbb {R}}_+$ is called regularly varying at $+\infty$ with index $\alpha$ if the following condition is satisfied:

$$\begin{aligned} \text {for all} \quad \lambda \in {\mathbb {R}}_+ \qquad \text {we have} \qquad \lim _{x\rightarrow +\infty } \frac{g(\lambda x)}{g(x)} = \lambda ^{\alpha }. \end{aligned}$$

When $\alpha = 0$ the function g is called slowly varying at $+\infty$.

A measurable function $g:[-a,0) \rightarrow {\mathbb {R}}_-$ is called regularly varying at 0 from the left with index $\alpha$ if the function $f(x) = -g(-x)$ where $x \in (0,a]$ is regularly varying at 0 from the right with index $\alpha$. When $\alpha = 0$ the function g is called slowly varying at 0 from the left.

We will often use “at $0_+$” as an abbreviation for the phrase “at 0 from the right” and “at $0_-$” as an abbreviation for the phrase “at 0 from the left.”

The Karamata’s theory of regular variation is commonly presented for functions regularly varying at $+\infty$. The results for functions regularly varying at $0_+$ follow from the following equivalence. Let f and g be measurable functions such that $g(x) = f(1/x)$ for all x in the domain of g for which 1/x is in the domain of f. Then g is regularly varying at $+\infty$ with index $\alpha$ if and only if f is regularly varying at $0_+$ with index $-\alpha$.

In this section some results will be presented at $0_+$ and some at $+\infty$. This choice is sometimes made based on our needs in this paper and sometimes on convenience.

Slow variation plays the central role in the theory of regular variation. That centrality is expressed in the following proposition that follows immediately from the definition.

Proposition A.2

Let $a, \alpha \in {\mathbb {R}}$ with $a > 0$ and let $f, g:(0,a]\rightarrow {\mathbb {R}}_+$ be measurable functions such that $g(x) = x^\alpha f(x)$ for all $x\in (0,a]$. The function g is regularly varying at $0_+$ with index $\alpha$ if and only if f is slowly varying at $0_+$.

The next theorem is Karamata’s Representation Theorem, see [12, Theorem 1.3.1] or [64] for Karamata’s original paper.

Theorem A.3

Let $a \in {\mathbb {R}}$. A function $f:[a,+\infty )\rightarrow {\mathbb {R}}_+$ is slowly varying at $+\infty$ if and only if there exist $b \in [a,+\infty )$, a measurable function $m: [b,+\infty ) \rightarrow {\mathbb {R}}_+$ and a continuous function $\varepsilon : [b,+\infty ) \rightarrow {\mathbb {R}}$ such that

$$\begin{aligned} \lim _{x\rightarrow +\infty } m(x) = M \in {\mathbb {R}}_+, \qquad \lim _{x\rightarrow +\infty } \varepsilon (x) = 0, \end{aligned}$$

and for all $x \ge b$ we have

$$\begin{aligned} f(x) = m(x) \exp \left( \int _{b}^{x} \frac{\varepsilon (t)}{t} dt \right) . \end{aligned}$$

The following property of regularly varying functions follows from Proposition A.2 and Theorem A.3, see [87, 1$^\circ$ on page 18].

Corollary A.4

If g is a regularly varying function at $+\infty$ with a positive (negative, respectively) index, then

$$\begin{aligned} \lim _{x\rightarrow +\infty } g(x) = +\infty \qquad \text {(} \lim _{x\rightarrow +\infty } g(x) = 0, \ \text {respectively)}. \end{aligned}$$

If f is a regularly varying function at $0_+$ with a positive (negative, respectively) index, then

$$\begin{aligned} \lim _{x\downarrow 0} f(x) = 0 \qquad \text {(} \lim _{x\downarrow 0} f(x) = +\infty , \ \text {respectively)}. \end{aligned}$$

1.2 Karamata’s Characterization and Consequences

The following theorem is our restatement of Karamata’s characterization of regular variation as it appears in [46, Theorem 1.2.1], [67, Theorems IV.5.2 and IV.5.3], [12, Theorems 1.5.11 and 1.6.1] and [14]. In [12, 14, 46, 67] regular variation at $+\infty$ is considered. Here we characterize regular variation at $0_+$.

Theorem A.5

Let $a\in {\mathbb {R}}_+$ and let $f:(0,a] \rightarrow {\mathbb {R}}_+$ be a locally integrable function on (0, a]. Let $\alpha , \gamma \in {\mathbb {R}}$ be such that $\gamma + \alpha \ne 0$ and consider the following two conditions:

$$\begin{aligned}{} & {} \int _{0}^{a} s^{\gamma -1} f(s) ds \quad \text {exists as an improper integral at }0, \end{aligned}$$

(A.1)

$$\begin{aligned}{} & {} \lim _{v \downarrow 0} \frac{1}{v^\gamma f(v)} \int _{0}^{v} s^{\gamma -1} f(s) ds = \frac{1}{\gamma + \alpha }. \end{aligned}$$

(A.2)

The following statements are equivalent:

• (a) f is regularly varying at $0_+$ with index $\alpha$.

• (b) For all $\gamma \in {\mathbb {R}}$ such that $\gamma + \alpha > 0$ conditions (A.1) and (A.2) hold.

• (c) There exists $\gamma \in {\mathbb {R}}$ such that $\gamma + \alpha > 0$ and (A.1) and (A.2) hold.

The next theorem is a reformulation of the preceding one in terms of the differential of the function under consideration.

Theorem A.6

Let $a, \hspace{1.111pt}\alpha , \hspace{1.111pt}\gamma \in {\mathbb {R}}$ be such that $a > 0, \hspace{1.111pt}\gamma \ne 0\hspace{1.111pt}$ and $\hspace{1.111pt}\gamma + \alpha \ne 0$. Let ${f:(0,a] \rightarrow {\mathbb {R}}_+}$ be a measurable function which is of bounded variation on each closed interval contained in (0, a]. Consider the following three conditions:

$$\begin{aligned}{} & {} \int _{0}^{a} s^\gamma df(s) \quad \text {exists as an improper Riemann-Stieltjes integral at }0, \nonumber \\ \end{aligned}$$

(A.3)

$$\begin{aligned}{} & {} \lim _{v \downarrow 0} v^\gamma f(v) = 0, \end{aligned}$$

(A.4)

$$\begin{aligned}{} & {} \lim _{v \downarrow 0} \frac{1}{v^\gamma f(v)} \displaystyle \int _{0}^{v}\! s^\gamma df(s) = \frac{\alpha }{\gamma + \alpha }. \end{aligned}$$

(A.5)

The following statements are equivalent:

• (i) f is regularly varying at $0_+$ with index $\alpha$.

• (ii) For all $\gamma \in {\mathbb {R}}\!\setminus \!\{0\}$ such that $\gamma + \alpha > 0$ conditions (A.3), (A.4) and (A.5) hold.

• (iii) There exists $\gamma \in {\mathbb {R}}\!\setminus \!\{0\}$ such that $\gamma + \alpha > 0$ and conditions (A.3), (A.4) and (A.5) hold.

Proof

Let $u, v \in (0,a]$ such that $u < v$. First notice that since f is of bounded variation on [u, v], see [96, Theorems 2.21 and 2.24], the integration by parts yields

$$\begin{aligned} \int _{u}^{v} s^\gamma df(s) = v^{\gamma } f(v) - u^{\gamma } f(u) - \gamma \int _{u}^{v} s^{\gamma -1} f(s) ds. \end{aligned}$$

(A.6)

Assume (i). Let $\gamma \in {\mathbb {R}}{\setminus }\{0\}$ be such that $\gamma + \alpha > 0$. Since by Definition A.1 the function $x\mapsto x^\gamma f(x)$ is regularly varying at $0_+$ with index $\gamma +\alpha$, Corollary A.4 yields (A.4).

Theorem A.5 implies that (A.1) and (A.2) hold. Letting $u\downarrow 0$ in (A.6) and using (A.1) yields (A.3) and

$$\begin{aligned} \frac{1}{v^{\gamma } f(v)} \int _{0}^{v} s^\gamma df(s) = 1 - \frac{\gamma }{v^{\gamma } f(v)} \int _{0}^{v} s^{\gamma -1} f(s) ds. \end{aligned}$$

(A.7)

Now letting $v\downarrow 0$ and using (A.2) we deduce (A.5), proving (ii).

The fact that (ii) implies (iii) is trivial. Now assume (iii). Letting $u\downarrow 0$ in (A.6) and using (A.3) yields (A.1), and we again deduce (A.7). Together (A.7) and (A.5) imply (A.2) in Theorem A.5. Thus, (c) in Theorem A.5 holds and (i) follows from Theorem A.5. $\square$

Let $\gamma > 0$. With the substitution $t = v^\gamma$, conditions (A.3), (A.4) and (A.5) can be rewritten as (see [79, Theorem 12.11] for the change of variables formula in Riemann-Stieltjes integral)

$$\begin{aligned}{} & {} \int _{0}^{a^\gamma } t\hspace{0.83328pt}df(t^{1/\gamma }) \quad \text {exists as an improper Riemann-Stieltjes integral at }0, \\{} & {} \lim _{t \downarrow 0} t f(t^{1/\gamma }) = 0, \\{} & {} \lim _{t \downarrow 0} \frac{1}{t f(t^{1/\gamma })} \displaystyle \int _{0}^{t} s\hspace{0.83328pt}df\bigl (s^{1/\gamma }\bigr ) = \frac{\alpha /\gamma }{1 + \alpha /\gamma }. \end{aligned}$$

This observation and Theorem A.6 (with $\gamma$ being 1 and $\alpha$ being $\alpha /\gamma$) yield the following equivalence: The function $t\mapsto f(t^{1/\gamma })$ with $t\in (0,a^\gamma ]$ is regularly varying at $0_+$ with index $\alpha /\gamma > -1$ if and only if conditions (A.3), (A.4), (A.5) hold. Here it is convenient to read the last fraction in (A.5) as $(\alpha /\gamma )/\bigl (1+(\alpha /\gamma )\bigr )$.

The next corollary generalizes the preceding equivalence to any increasing bijection on [0, a].

Corollary A.7

Let $\alpha , a, b \in {\mathbb {R}}$ be such that $a, b > 0$ and $\alpha > -1$. Let ${f:(0,b] \rightarrow {\mathbb {R}}_+}$ be a function of bounded variation on every closed subinterval of (0, b] and let $g:[0,b] \rightarrow [0,a]$ be an increasing bijection. The function is regularly varying at $0_+$ with index $\alpha > -1$ if and only if the following three conditions are satisfied:

$$\begin{aligned}{} & {} \int _{0}^{b}\! g(s)\hspace{0.83328pt}df(s) \quad \text {exists as an improper Riemann-Stieltjes integral at }0,\nonumber \\ \end{aligned}$$

(A.8)

$$\begin{aligned}{} & {} \lim _{v \downarrow 0} f(v) g(v) = 0, \end{aligned}$$

(A.9)

$$\begin{aligned}{} & {} \lim _{v \downarrow 0} \frac{1}{f(v)g(v)}\! \int _{0}^{v}\! g(s)\hspace{0.83328pt}df(s) = \frac{\alpha }{1+\alpha }. \end{aligned}$$

(A.10)

Proof

Let $u, v \in (0,b]$ such that $u < v$. As in the preceding theorem we notice that since f is of bounded variation on [u, v] the integration by parts ( [96, Theorem 2.21]) yields

$$\begin{aligned} \int _{u}^{v} g(s)\hspace{0.83328pt}df(s) = f(v)g(v) - f(u)g(u) - \int _{u}^{v} f(s)\hspace{0.83328pt}dg(s). \end{aligned}$$

(A.11)

In this proof we will also use that, since g is a continuous increasing bijection, we have that $u \downarrow 0$ if and only if $g(u)\downarrow 0$.

Assume (A.8), (A.9) and (A.10). Letting $u\downarrow 0$ and using (A.8) and (A.9) in (A.11) yields

$$\begin{aligned} \int _{0}^{v} g(s)\hspace{0.83328pt}df(s) = f(v)g(v) - \int _{0}^{v} f(s)\hspace{0.83328pt}dg(s) \end{aligned}$$

(A.12)

for all $v \in (0,b]$. Therefore, for all $v \in (0,b]$ we have

$$\begin{aligned} \begin{aligned} \frac{1}{f(v)g(v)} \int _{0}^{v} g(s)\hspace{0.83328pt}df(s)&= 1 - \frac{1}{f(v)g(v)}\int _{0}^{v} f(s) dg(s) \\&= 1 - \frac{1}{f(v)g(v)}\int _{0}^{g(v)} f\bigl (g^{-1}(t)\bigr ) dt, \end{aligned} \end{aligned}$$

(A.13)

where, for the second equality, we used the change of variables formula in Riemann-Stieltjes integral, [79, Theorem 12.11]. Now (A.10) implies

$$\begin{aligned} \frac{1}{1+\alpha } = \lim _{v \downarrow 0} \frac{1}{f(v)g(v)}\int _{0}^{g(v)} f\bigl (g^{-1}(t)\bigr ) dt = \lim _{u \downarrow 0} \frac{1}{u f\bigl (g^{-1}(u)\bigr )}\int _{0}^{u} f\bigl (g^{-1}(t)\bigr ) dt. \end{aligned}$$

Since we assume $1+\alpha > 0$, Theorem A.5 yields that is regularly varying at $0_+$ with index $\alpha$.

To prove the converse assume that is regularly varying at $0_+$ with index $\alpha > -1$. Then the function $x\mapsto xf\bigl (g^{-1}(x)\bigr )$ is regularly varying at $0_+$ with index $\alpha + 1 > 0$ and (A.9) follows from Corollary A.4 after a change of variables in the limit. By the change of variables formula for all $u \in (0,a]$ we have

$$\begin{aligned} \int _{u}^{a} s\hspace{0.83328pt}d f\bigl (g^{-1}(s)\bigr ) = \int _{g^{-1}(u)}^{b} g(t)\hspace{0.83328pt}df(t). \end{aligned}$$

Consequently, (A.8) follows from (A.3) in Theorem A.6 applied to with $\gamma = 1$. Therefore, (A.12) and consequently (A.13) both hold. Now (A.10) follows from (A.2) in Theorem A.5 with $\gamma =1$. $\square$

1.3 Asymptotic Equivalence of Functions on a Sequence

In the next definition we extend the notation $\sim$ of asymptotic equivalence of functions to hold only on a sequence.

Definition A.8

Let $a \in {\mathbb {R}}_+$. For functions $f,g:[a,+\infty ) \rightarrow {\mathbb {R}}_+$ we write

if and only if there exists an increasing sequence $(x_n)$ in $[a,+\infty )$ such that

$$\begin{aligned} \lim _{n\rightarrow +\infty } x_n = +\infty \quad \text {and} \quad \lim _{n\rightarrow +\infty } \frac{f(x_n)}{g(x_n)} = 1. \end{aligned}$$

For functions $f,g:(0,a] \rightarrow {\mathbb {R}}_+$ we write

if and only if there exists a decreasing sequence $(x_n)$ in (0, a] such that

$$\begin{aligned} \lim _{n\rightarrow +\infty } x_n = 0 \quad \text {and} \quad \lim _{n\rightarrow +\infty } \frac{f(x_n)}{g(x_n)} = 1. \end{aligned}$$

Recall, see [7, 91, 5.10.11], that for a function $\phi :[a,+\infty )$ a real number L is a cluster value of $\phi$ at $+\infty$ if for every $\epsilon > 0$ and for every $X \in {\mathbb {R}}$ there exists $x > X$ such that $|\phi (x) - L| < \epsilon$. Similarly, for a function $\phi :(0, a]$ a real number L is a cluster value of $\phi$ at $0_+$ if for every $\epsilon > 0$ and for every $\delta > 0$ there exists $x \in (0,\delta )$ such that $|\phi (x) - L| < \epsilon$. Notice that at $+\infty$ (at $0_+$) if and only if 1 is a cluster value of the function f/g at $+\infty$ (at $0_+$).

Proposition A.9

Let f and g be regularly varying functions at $+\infty$ with indices $\alpha$ and $\beta$, respectively. If at $+\infty$, then $\alpha = \beta$.

Proof

We will prove the contrapositive. Assume that $\alpha < \beta$. Since the function f(x)/g(x) is regularly varying with index $\alpha -\beta <0$ it follows from Corollary A.4 that $\lim _{x\rightarrow +\infty }f(x)/g(x)=0$. Thus, at $+\infty$ is not true. If $\alpha > \beta$ the preceding limit is $+\infty$, so at $+\infty$ is not true in this case either. $\square$

The converse of the preceding proposition is not true. For example, let f be a slowly varying function at $+\infty$ and $g=2f$. Then $\alpha = \beta =0$, but at $+\infty$ is clearly not true.

The following theorem extends [84, Proposition 0.8(vi)] to the concept introduced in the previous definition. This theorem can be deduced from [15, Corollary 7.66]. A direct proof is presented in [19, Theorem A.11, Corollary A.12].

Theorem A.10

Let f and g be strictly monotonic positive functions defined in a neighbourhood of $0_+$ and let f be regularly varying at $0_+$ with a nonzero index.

• (a) If f and g are increasing with 0 limit at $0_+$, then the inverses $f^{-1}$ and $g^{-1}$ are also increasing, defined in a neighbourhood of $0_+$ and the following equivalence holds

  

• (b) If f and g are decreasing and unbounded, then the inverses $f^{-1}$ and $g^{-1}$ are decreasing, defined in a neighbourhood of $+\infty$ and the following equivalence holds

  

The following corollary is a consequence of the fact that the negation of at $0_+$ is the statement

$$\begin{aligned} \left( \frac{f(x)}{g(x)} - 1 \right) ^{-1} = O(1) \quad \text {as} \quad x \downarrow 0. \end{aligned}$$

Each of the two statements in Theorem A.10 can be expressed using one of these negations. We state only the analogue of the last statement in Theorem A.10 since that is what is used in Theorem 4.16.

Corollary A.11

Let f and g be strictly monotonic positive functions defined in a neighbourhood of $0_+$ and let f be regularly varying at $0_+$ with a nonzero index. If f and g are decreasing and unbounded, then the inverses $f^{-1}$ and $g^{-1}$ are decreasing, defined in a neighbourhood of $+\infty$ and the following equivalence holds

$$\begin{aligned} \left( \frac{f(x)}{g(x)} - 1 \right) ^{\!\!-1} = O(1) \ \text {as} \ x \downarrow 0 \quad \Leftrightarrow \quad \left( \frac{f^{-1}(y)}{g^{-1}(y)} - 1 \right) ^{\!\!-1} = O(1) \ \text {as} \ y \rightarrow +\infty . \end{aligned}$$

Clearly $f\sim g$ at $+\infty$ implies at $+\infty$. In the next example we will demonstrate that at $+\infty$ does not imply $f\sim g$ at $+\infty$ even for smooth normalized slowly varying increasing functions f and g for which f/g is normalized slowly varying function.

1.4 Positively Increasing Functions

The following class of functions was introduced as a generalization of regularly varying functions with positive index, see [15, Section 3.1 and Definition 3.26].

Definition A.12

Let $a \in {\mathbb {R}}_+$. A nondecreasing function $f: (0,a] \rightarrow {\mathbb {R}}_+$ is called positively increasing at 0 from the right if there exists $\lambda \in (0,1)$ such that

$$\begin{aligned} \limsup _{x\downarrow 0}\frac{f(\lambda x)}{f(x)} < 1. \end{aligned}$$

A function $g:[-a,0) \rightarrow {\mathbb {R}}_-$ is called positively increasing at 0 from the left if the function $f(x) = -g(-x), x \in [-a,0)$, is positively increasing at 0 from the right.

A function $g:[a,+\infty ) \rightarrow {\mathbb {R}}_+$ is called positively increasing at $+\infty$ if the function $f(x) = 1/g(1/x), x \in (0,1/a]$, is positively increasing at 0 from the right. A function $g:(-\infty ,-a] \rightarrow {\mathbb {R}}_-$ is called positively increasing at $-\infty$ if the function $f(x) = -1/g(-1/x), x \in (0,1/a]$, is positively increasing at 0 from the right.

The relationship between regularly varying and positively increasing functions at $+\infty$, and analogously at $-\infty$, $0_+$ and $0_-$, is as follows. Each regularly varying function with positive index is positively increasing, while a regularly varying function with a nonpositive index is not positively increasing. In particular, a slowly varying function is not positively increasing. The exponential function $\exp$ is positively increasing at $+\infty$ but not regularly varying at $+\infty$. As was shown in [19, Example A.17], there exists a nondecreasing function $f:[1,+\infty )\rightarrow {\mathbb {R}}_+$ which is neither positively increasing nor slowly varying at $+\infty$.