On a property of norming constants of Sturm-Liouville problem

TL;DR

This paper explores how the norming constants in Sturm-Liouville problems depend on boundary conditions, using the Gelfand-Levitan method to establish a new connection in inverse problem solutions.

Contribution

It introduces a novel relationship between norming constants and boundary conditions in Sturm-Liouville problems via the Gelfand-Levitan approach.

Findings

Established a dependence of norming constants on boundary conditions.

Applied Gelfand-Levitan method to inverse Sturm-Liouville problems.

Provided insights into inverse spectral problem solutions.

Abstract

A connection, which shows the dependence of norming constants on boundary conditions, was found using the Gelfand-Levitan method for the solution of inverse Sturm-Liouville problem.

Equations

$$\ell y\equiv -y^{\prime \prime }+q(x)y=\mu y,x\in (0,\pi ),\mu \in \mathbb{C},$$

$$y(0)\cos \alpha +y^{\prime }(0)\sin \alpha =0,\alpha \in (0,\pi ),$$

$$y(\pi )\cos \beta +y^{\prime }(\pi )\sin \beta =0,\beta \in (0,\pi ),$$

$$\phi (0,\mu ,\alpha ,q)=\sin \alpha ,{\phi }^{\prime }(0,\mu ,\alpha ,q)=-\cos \alpha ,$$

$$\psi (\pi ,\mu ,\beta ,q)=\sin \beta ,{\psi }^{\prime }(\pi ,\mu ,\beta ,q)=-\cos \beta ,$$

$$\Phi (\mu )=\Phi (\mu ,\alpha ,\beta )\stackrel{def}{=}\phi (\pi ,\mu ,\alpha )\cos \beta +{\phi }^{\prime }(\pi ,\mu ,\alpha )\sin \beta =0,$$

$$\Psi (\mu )=\Psi (\mu ,\alpha ,\beta )\stackrel{def}{=}\psi (0,\mu ,\beta )\cos \alpha +{\psi }^{\prime }(0,\mu ,\beta )\sin \alpha =0.$$

$$W(x)=W(x,\phi ,\psi )=\phi \cdot {\psi }^{\prime }-{\phi }^{\prime }\psi$$

$$\phi (x,{\mu }_n)=c_n\cdot \psi (x,{\mu }_n).$$

$$a_n=a_n(q,\alpha ,\beta )=\underset{0}{\overset{\pi }{\int }}|{\phi }_n(x){|}^{2}dx,n=0,1,2,\dots ,$$

$$b_n=b_n(q,\alpha ,\beta )=\underset{0}{\overset{\pi }{\int }}|{\psi }_n(x){|}^{2}dx,n=0,1,2,\dots$$

$$\tilde{\phi }(0,\mu ,\alpha ,q)=1,\tilde{\phi }(x,\mu ,\alpha ,q)=-\cot \alpha ,$$

$${\tilde{b}}_n=\frac{b_n}{{\sin }^2\beta }=\frac{a_n}{c_n^2{\sin }^2\beta }=\frac{{\tilde{a}}_n{\sin }^2\alpha }{c_n^2{\sin }^2\beta }$$

$$\frac{1}{{\tilde{a}}_0}-\frac{1}{\pi }+\sum _{n=1}^{\infty }\left(\frac{1}{{\tilde{a}}_n}-\frac{2}{\pi }\right)=\cot \alpha ,$$

$$\frac{1}{{\tilde{b}}_0}-\frac{1}{\pi }+\sum _{n=1}^{\infty }\left(\frac{1}{{\tilde{b}}_n}-\frac{2}{\pi }\right)=-\cot \beta .$$

$$\tilde{\phi }(x,\lambda ,\alpha ,q)=\cos \lambda x+\underset{0}{\overset{x}{\int }}G(x,t)\cos \lambda tdt,$$

$$G(x,x)=-\cot \alpha +\frac{1}{2}\underset{0}{\overset{x}{\int }}q(s)ds.$$

$$G(x,t)+F(x,t)+\underset{0}{\overset{x}{\int }}G(x,s)F(s,t)ds=0,0\le t\le x,$$

$$F(x,t)=\sum _{n=0}^{\infty }\left(\frac{\cos {\lambda }_{n}x\cos {\lambda }_{n}t}{{\tilde{a}}_n}-\frac{\cos nx\cos nt}{a_n^0}\right),$$

$$G(0,0)=-F(0,0)=-\sum _{n=0}^{\infty }\left(\frac{1}{{\tilde{a}}_n}-\frac{1}{a_n^0}\right)=$$

$$=-\left(\frac{1}{{\tilde{a}}_0}-\frac{1}{\pi }\right)-\sum _{n=1}^{\infty }\left(\frac{1}{{\tilde{a}}_n}-\frac{2}{\pi }\right)=-\cot \alpha .$$

$$p(x,{\mu }_n)=\frac{\phi (\pi -x,{\mu }_n,\alpha ,q)}{\phi (\pi ,{\mu }_n,\alpha ,q)}=\frac{\phi (\pi -x,{\mu }_n)}{\phi (\pi ,{\mu }_n)}.$$

$$p^{\prime }(x,{\mu }_n)=-\frac{{\phi }^{\prime }(\pi -x,{\mu }_n)}{\phi (\pi ,{\mu }_n)},p^{\prime \prime }(x,{\mu }_n)=\frac{{\phi }^{\prime \prime }(\pi -x,{\mu }_n)}{\phi (\pi ,{\mu }_n)},$$

$$-p^{\prime \prime }(x,{\mu }_n)+q(\pi -x)p(x,{\mu }_n)={\mu }_{n}p(x,{\mu }_n)$$

$$p(0,{\mu }_n)=1,p^{\prime }(0,{\mu }_n)=-\frac{{\phi }^{\prime }(\pi ,{\mu }_n)}{\phi (\pi ,{\mu }_n)}=-(-\cot \beta )=\cot \beta =-\cot (\pi -\beta ).$$

$$p(\pi ,{\mu }_n)=\frac{\phi (0,{\mu }_n)}{\phi (\pi ,{\mu }_n)}=\frac{\sin \alpha }{\phi (\pi ,{\mu }_n)}=\frac{\sin (\pi -\alpha )}{\phi (\pi ,{\mu }_n)},$$

$$p^{\prime }(\pi ,{\mu }_n)=-\frac{{\phi }^{\prime }(0,{\mu }_n)}{\phi (\pi ,{\mu }_n)}=-\frac{-\cos \alpha }{\phi (\pi ,{\mu }_n)}=\frac{-\cos (\pi -\alpha )}{\phi (\pi ,{\mu }_n)}.$$

$$p_n(\pi )\cos (\pi -\alpha )+p_n^{\prime }(\pi )\sin (\pi -\alpha )=0,n=0,1,2,\dots$$

$$\left(\frac{1}{{\hat{a}}_0}-\frac{1}{\pi }\right)+\sum _{n=1}^{\infty }\left(\frac{1}{{\hat{a}}_n}-\frac{2}{\pi }\right)=\cot (\pi -\beta )=-\cot \beta .$$

$${\hat{a}}_n=\underset{0}{\overset{\pi }{\int }}{p}^2(x,{\mu }_n)dx=\underset{0}{\overset{\pi }{\int }}\frac{{\phi }^2(\pi -x,{\mu }_n)}{{\phi }^2(\pi ,{\mu }_n)}dx=$$

$$=-\frac{1}{{\phi }^2(\pi ,{\mu }_n)}\underset{\pi }{\overset{0}{\int }}{\phi }^2(s,{\mu }_n)ds=\frac{1}{{\phi }^2(\pi ,{\mu }_n)}\underset{0}{\overset{\pi }{\int }}{\phi }^2(s,{\mu }_n)ds=$$

$$=\frac{a_n(q,\alpha ,\beta )}{{\phi }^2(\pi ,{\mu }_n)}=\frac{{\tilde{a}}_n{\sin }^2\alpha }{c_n^2{\sin }^2\beta }={\tilde{b}}_n.$$

Full text

MSC2010: 34B24, 34L20. Keywords:   Sturm–Liouville problem, eigenvalues, norming constants.

1 Introduction.

Let $L(q,\alpha,\beta)$ denote the Sturm–Liouville boundary value problem

[TABLE]

where $q$ is a real-valued, summable on $[0,\pi]$ function (we write $q\in L^{1}_{{\mathbb{R}}}[0,\pi]$). By $L(q,\alpha,\ \beta)$ we  also  denote  the  self-adjoint  operator,  generated  by  the  problem (1)–(3) (see [2]). It is known that under these conditions the spectra of the operator $L(q,\alpha,\beta)$ is discrete and consists of real, simple eigenvalues [2], which we denote by $\mu_{n}=\mu_{n}(q,\alpha,\beta)=\lambda_{n}^{2}(q,\alpha,\beta)$, $n=0,1,2,\ldots$, emphasizing the dependence of $\mu_{n}$ on $q$, $\alpha$ and $\beta$.

Let $\varphi(x,\mu,\alpha,q)$ and $\psi(x,\mu,\beta,q)$ are the solutions of Eq. (1), which satisfy the initial conditions

[TABLE]

[TABLE]

correspondingly. The eigenvalues $\mu_{n}=\mu_{n}(q,\alpha,\beta)$, $n=0,1,2,\ldots$, of $L(q,\alpha,\beta)$ are the solutions of the equation

[TABLE]

or the equation

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According to the well-known Liouville formula, the wronskian

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of the solutions $\varphi$ and $\psi$ is constant. It follows that $W(0)=W(\pi)$ and consequently $\Psi(\mu,\alpha,\beta)=-\Phi(\mu,\alpha,\beta).$ It is easy to see that the functions $\varphi_{n}(x)=\varphi(x,\mu_{n},\alpha)$ and $\psi_{n}(x)=\psi(x,\mu_{n},\beta)$, $n=0,1,2,\ldots$, are the eigenfunctions, corresponding to the eigenvalue $\mu_{n}$. Since all eigenvalues are simple, there exist constants $c_{n}=c_{n}(q,\alpha,\beta)$, $n=0,1,2,\ldots$, such that

[TABLE]

The squares of the $L^{2}$-norm of these eigenfunctions:

[TABLE]

[TABLE]

are called the norming constants.

In this paper we consider the case $\alpha,\beta\in(0,\pi)$, i.e. we assume that $\sin\alpha\neq 0$ and $\sin\beta\neq 0$. In this case we consider the solution $\tilde{\varphi}(x,\mu,\alpha,q):=\dfrac{\varphi(x,\mu,\alpha,q)}{\sin\alpha}$ of the equation (1) which has the initial values

[TABLE]

and also we consider the solution $\tilde{\psi}(x,\mu,\beta,q):=\dfrac{\psi(x,\mu,\beta,q)}{\sin\beta}$. Of course, the functions $\tilde{\varphi}_{n}(x):=\tilde{\varphi}(x,\mu_{n},\alpha,q)$ and $\tilde{\psi}_{n}(x):=\tilde{\psi}(x,\mu_{n},\alpha,q)$, $n=0,1,2,\ldots,$ are the eigenfunctions, corresponding to the eigenvalue $\mu_{n}$. It follows from (4) that for norming constants $\tilde{a}_{n}:=\|\tilde{\varphi}_{n}\|^{2}=\dfrac{a_{n}}{\sin^{2}\alpha},$ $\tilde{b}_{n}:=\|\tilde{\psi}_{n}\|^{2}=\dfrac{b_{n}}{\sin^{2}\beta}$ the following connections

[TABLE]

hold.

2 The Main Result.

The aim of this paper is to prove the following assertion.

Theorem. For the norming constants $\tilde{a}_{n}$ and $\tilde{b}_{n}$ the following connections hold:

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For the solution $\tilde{\varphi}$ it is well known the representation (see [3, 4])

[TABLE]

where for the kernel $G(x,t)$ we have (in particular) (see [4])

[TABLE]

Besides, it is known that $G(x,t)$ satisfies to the Gelfand–Levitan integral equation

[TABLE]

where the function $F(x,t)$ is defined by the formula (see [4])

[TABLE]

where $a_{0}^{0}=\pi$ and $a_{n}^{0}={\pi}/2$ for $n=1,2,\ldots$ It easily follows from (9)–(11) that

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[TABLE]

Thus, (6) is proved.

Let us now consider the functions ($n=0,1,2,\ldots$)

[TABLE]

Since $\varphi(x,\mu,\alpha,q)$ satisfies the Eq. (1) and

[TABLE]

we can see, that $p(x,\mu_{n})$ satisfies the equation

[TABLE]

and the initial conditions

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We also have

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[TABLE]

From this it follows that $p_{n}(x):=p(x,\mu_{n})$ satisfies the boundary condition

[TABLE]

Let us denote $q^{*}(x):=q(\pi-x)$. Since $\mu_{n}(q^{*},\pi-\beta,\pi-\alpha)=\mu_{n}(q,\alpha,\beta)$ (it is easy to prove and is well known, see for example [5]), it follows, that $p_{n}(x),\ n=0,1,2,\ldots,$ are the eigenfunctions of the problem $L(q^{*},\pi-\beta,\pi-\alpha)$, which have the initial conditions (14), i.e. $p_{n}(x)=\tilde{\varphi}(x,\mu_{n},\pi-\beta,q^{*}),\ n=0,1,2,\ldots$

Thus, as in (12), for the norming constants $\hat{a}_{n}=\|p(\cdot,\mu_{n})\|^{2}$ we have

[TABLE]

On the other hand, for the norming constants $\hat{a}_{n}$, according to (4), (5) and (13), we have

[TABLE]

[TABLE]

[TABLE]

Therefore, we can rewrite (15) in the form

[TABLE]

Thus (7) is true and Theorem is proved. $\Box$

3 Remark.

It is known from the inverse Sturm–Liouville problems, that the set of eigenvalues \Big{\{}\mu_{n}\Big{\}}_{n=0}^{\infty} and the norming constants \Big{\{}\tilde{a}_{n}\Big{\}}_{n=0}^{\infty} uniquely determine the problem $L(q,\alpha,\beta)$. That means, in particular, that we can determine $\{\tilde{b}_{n}\}_{n=0}^{\infty}$ by these two sequences. Now we will derive the precise formulae for these connections.

It is known that the specification of the spectra \Big{\{}\mu_{n}(q,\alpha,\beta)\Big{\}}_{n=0}^{\infty} uniquely determines the characteristic function $\Phi(\mu)$ (see [5], Lemma 1($iii$); [6], Lemma 2.2) and also its derivative $\partial\Phi(\mu)/\partial\mu=\dot{\Phi}(\mu)$ ([6] Lemma 2.3).

In particular, if $\alpha,\beta\in(0,\pi)$ the following formulas hold:

[TABLE]

and (if $n\neq 0$, i.e. $n=1,2,\ldots$)

[TABLE]

On the other hand, it is easy to prove the relation (see [6], Eq. (2.16) in Lemma 2.2 and see [5], Lemma 1 ($iii$))

[TABLE]

Taking into account the connections (5) and (16)–(18) we can find formulae for $\dfrac{1}{\tilde{b}_{0}}$  and  $\dfrac{1}{\tilde{b}_{n}},\ n=1,2,\ldots:$

[TABLE]

[TABLE]

So, we can change the second assertion in Theorem by the following equation

[TABLE]

[TABLE]

This research is supported by the Open Society Foundations–Armenia, within the Education program, grant No 18742

REFERENCES