Sturm Liouville Problem

A classic Sturm-Liouville problem (after Charles-François Sturm (1803–1855) and Joseph Liouville (1809–1882)) is the following eigenvalue problem from analysis : Consider the differential equation of the 2nd order:

${\displaystyle -\left(p\cdot \psi '\right)'+q\cdot \psi =\lambda \cdot w\cdot \psi }$

where are coefficient functions. Find all complex numbers for which the differential equation on the interval has a solution that meets the boundary conditions ${\displaystyle p,q,w}$ ${\displaystyle \lambda }$ ${\displaystyle (a,b)}$

${\displaystyle {\begin{aligned}\cos(\alpha )\psi (a)&+\sin(\alpha )p(a)\psi '(a)=0\\\cos(\beta )\psi (b)&+\sin(\beta )p(b)\psi '(b)=0\end{aligned}}}$

is sufficient ( ). ${\displaystyle \alpha ,\beta \in [0,\pi )}$

One uses the linear operator of the form

${\displaystyle {\mathcal {L}}={\frac {1}{w}}\left(-{\frac {\mathrm {d} }{\mathrm {d} x}}\,p\,{\frac {\mathrm {d} }{\mathrm {d} x}}+q\right)}$

one, the Sturm-Liouville operator , the eigenvalue equation can be treated with the help of methods from functional analysis ( spectral theory ) in the Hilbert space of functions that can be square-integrated with respect to the weight function. ${\displaystyle {\mathcal {L}}\psi =\lambda \psi }$ ${\displaystyle w}$

If the interval is compact and the coefficient functions can be integrated, one speaks of a regular Sturm-Liouville problem. If the interval is unlimited or if the coefficient functions can only be integrated locally, one speaks of a singular Sturm-Liouville problem. ${\displaystyle w,p^{-1},q}$

Regular Storm Liouville problems

The eigenvalue equation

${\displaystyle -(p\cdot \psi ')'+q\cdot \psi =\lambda \cdot w\cdot \psi }$

with integrable real functions , together with boundary conditions of the form ${\displaystyle w(x)>0,p(x)^{-1}>0,q(x)}$

${\displaystyle \cos(\alpha )\psi (a)+\sin(\alpha )p(a)\psi '(a)=0,\quad \cos(\beta )\psi (b)+\sin(\beta )p(b)\psi '(b)=0,\qquad \alpha ,\beta \in [0,\pi ),}$

is called a regular Sturm-Liouville problem over the interval if this interval is finite. ${\displaystyle [a,b]}$

In the case one speaks of Dirichlet boundary conditions and in the case of Neumann boundary conditions , whereby the existence and uniqueness of the solution with the boundary conditions is ensured. ${\displaystyle \psi (a)=\psi (b)=0}$ ${\displaystyle \psi '(a)=\psi '(b)=0}$

For the regular Sturm-Liouville problem it holds that there is a countable sequence of real eigenvalues ​​that diverges towards : ${\displaystyle +\infty }$

${\displaystyle \lambda _{1}<\lambda _{2}<\lambda _{3}<\cdots <\lambda _{n}<\cdots \to \infty .}$

The eigenvalues ​​behave asymptotically (Weyl asymptotics) like

${\displaystyle \lambda _{n}=\pi ^{2}\left(\int _{a}^{b}{\sqrt {\frac {w(x)}{p(x)}}}\mathrm {d} x\right)^{-2}n^{2}+O(n).}$

The associated eigenfunctions form an orthonormal basis in the Hilbert space of the square-integrable functions with respect to the weight function. ${\displaystyle \psi _{n}}$ ${\displaystyle L^{2}([a,b],w(x)\mathrm {d} x)}$ ${\displaystyle w}$

properties

For the regular Sturm-Liouville problem one is interested in describing the behavior of the eigenfunctions without having precise knowledge of them. In this respect, the following sentences, which partly go back to Charles-François Sturm, give an overview of the properties of the solutions to the Sturm-Liouville problem.

For this purpose, the homogeneous differential equation for is considered and the following requirements are placed on the coefficient functions : ${\displaystyle \textstyle {\mathcal {L}}\psi ={\frac {1}{w}}\left(-{\frac {\mathrm {d} }{\mathrm {d} x}}\,p\,{\frac {\mathrm {d} }{\mathrm {d} x}}+q\right)\psi =0}$ ${\displaystyle w=1}$ ${\displaystyle p,q}$

• ${\displaystyle p\in C^{1}((a,b),\mathbb {R} )}$and , ${\displaystyle p>0}$
• ${\displaystyle q\in C^{0}((a,b),\mathbb {R} )}$and . ${\displaystyle q>0}$

Additional requirements are formulated in the corresponding sentences.

Amplitude theorem

Since the amplitudes indicate the absolute value of the local extreme values, the following sentence describes the behavior of the amplitudes of successive zeros.

Is a departure from the aforementioned conditions , monotonically increasing or monotonically decreasing, and at an appropriate interval is a non-trivial solution . For the amplitudes of two consecutive extreme points of : ${\displaystyle p,q\in C^{1}((a,b),\mathbb {R} )}$ ${\displaystyle p,q}$ ${\displaystyle (c,d)\subseteq (a,b)}$ ${\displaystyle \phi }$ ${\displaystyle {\mathcal {L}}\phi =0}$ ${\displaystyle c<x_{k}<x_{k+1}<d}$ ${\displaystyle \phi }$

${\displaystyle |\phi (x_{k+1})|\geq |\phi (x_{k})|{\text{ wenn }}(pq)'<0}$ and

${\displaystyle |\phi (x_{k+1})|\leq |\phi (x_{k})|{\text{ wenn }}(pq)'>0}$.

proof

It is a non-trivial solution and ${\displaystyle \phi }$

${\displaystyle \psi =\phi ^{2}+{\frac {1}{pq}}\left(p\phi '\right)^{2}}$.

There is no solution to the Sturm-Liouville differential equation, but a function that is equipped with the same extreme points and zeros as . With the help of this construction follows with the Sturm-Liouville differential equation ${\displaystyle \psi }$ ${\displaystyle \phi }$ ${\displaystyle (p\phi ')'=-q\phi }$

${\displaystyle {\begin{aligned}\psi '&=2\phi \phi '+{\frac {1}{pq}}2p\phi '\left(p\phi '\right)'-{\frac {(pq)'}{(pq)^{2}}}\left(p\phi '\right)^{2}\\&=2\phi \phi '-2\phi \phi '-{\frac {(pq)'}{(pq)^{2}}}\left(p\phi '\right)^{2}\\&=-(pq)'\left({\frac {\phi '}{q}}\right)^{2}.\end{aligned}}}$

If it is also taken into account that is at every extreme point , then applies to a with ${\displaystyle \phi '(x_{k+1})=\phi '(x_{k})=0}$ ${\displaystyle \xi }$ ${\displaystyle c<x_{k}\leq \xi \leq x_{k+1}<d}$

${\displaystyle {\begin{aligned}\psi '(\xi )\geq 0&{\text{ wenn }}(p(\xi )q(\xi ))'<0\\\psi '(\xi )\leq 0&{\text{ wenn }}(p(\xi )q(\xi ))'>0.\end{aligned}}}$

Hence the slope of is influenced by the value of the derivative of . Since the slope of on inherited, one obtains for the amount: ${\displaystyle \psi }$ ${\displaystyle (pq)'}$ ${\displaystyle \psi }$ ${\displaystyle \phi ^{2}}$

${\displaystyle |\phi (x_{k+1})|\geq |\phi (x_{k})|{\text{ wenn }}(pq)'<0}$ and

${\displaystyle |\phi (x_{k+1})|\leq |\phi (x_{k})|{\text{ wenn }}(pq)'>0}$.

${\displaystyle \Box }$

Oscillation set

The principle of oscillation states that, in addition to the requirements described above, the following also applies: ${\displaystyle {\mathcal {L}}\psi =0}$ ${\displaystyle p,q}$

${\displaystyle \lim _{b\to \infty }\int _{a}^{b}{\frac {1}{p(x)}}\mathrm {d} x}$and are divergent, ${\displaystyle \lim _{b\to \infty }\int _{a}^{b}q(x)\mathrm {d} x}$

then every nontrivial solution on the interval is oscillatory . ${\displaystyle (a,\infty )}$

In addition, in the case of Dirichlet boundary conditions, every th eigenfunction has exactly zeros in the interval . ${\displaystyle n}$ ${\displaystyle \psi _{n}}$ ${\displaystyle n-1}$ ${\displaystyle (a,b)}$

proof

Be like non-trivial solutions of the homogeneous differential equation. With and because is and thus: ${\displaystyle \phi }$ ${\displaystyle \psi :=p\phi '}$ ${\displaystyle \phi '={\tfrac {1}{p}}\psi }$ ${\displaystyle -(p\phi ')'+q\phi =0}$ ${\displaystyle \psi '=(p\phi ')'=q\phi }$

(1) . ${\displaystyle \quad {\begin{pmatrix}\phi \\\psi \end{pmatrix}}'={\begin{pmatrix}{\frac {1}{p}}\psi \\q\phi \end{pmatrix}}={\begin{pmatrix}0&{\frac {1}{p}}\\q&0\end{pmatrix}}{\begin{pmatrix}\phi \\\psi \end{pmatrix}}}$

This system of linear differential equations only has non-trivial solutions if it holds for each , since otherwise and therefore would have to be. ${\displaystyle \xi \geq a}$ ${\displaystyle {\Big (}{\begin{smallmatrix}\phi (\xi )\\\psi (\xi )\end{smallmatrix}}{\Big )}={\Big (}{\begin{smallmatrix}\phi (\xi )\\p\phi '(\xi )\end{smallmatrix}}{\Big )}\neq {\big (}{\begin{smallmatrix}0\\0\end{smallmatrix}}{\big )}}$ ${\displaystyle \phi (\xi )=\phi (\xi )'=0}$ ${\displaystyle \phi (\xi )\equiv 0}$

We are therefore looking for oscillatory solutions that are obtained by means of the Prüfer transformation in plane polar coordinates:

(2) . ${\displaystyle \quad {\begin{pmatrix}\phi (x)\\\psi (x)\end{pmatrix}}=\rho (x){\begin{pmatrix}\sin \vartheta (x)\\\cos \vartheta (x)\end{pmatrix}}}$

Where and the associated argument function is: ${\displaystyle \rho (x)={\big (}\phi ^{2}(x)+\psi ^{2}(x){\big )}^{1/2}}$

${\displaystyle \vartheta (x)=\arctan {\tfrac {\phi (x)}{\psi (x)}}\quad }$or . ${\displaystyle \quad \vartheta (x)=\operatorname {arccot} {\tfrac {\psi (x)}{\phi (x)}}}$

Claim: If , then just as we have infinitely many zeros. ${\displaystyle \lim _{x\to \infty }\vartheta (x)\to \infty }$ ${\displaystyle \sin \vartheta (x)}$ ${\displaystyle \phi (x)}$

Justification: From (1) and (2) follows

(3) and ${\displaystyle \quad \phi '\;{\stackrel {\mathrm {(2)} }{=}}{\big (}\rho \sin \vartheta {\big )}'=\rho '\sin \vartheta +\rho \vartheta '\cos \vartheta \;{\stackrel {\mathrm {(1)} }{=}}{\frac {\rho }{p}}\cos \vartheta }$

(4) . ${\displaystyle \quad \psi '\;{\stackrel {\mathrm {(2)} }{=}}{\big (}\rho \cos \vartheta {\big )}'=\rho '\cos \vartheta -\rho \vartheta '\sin \vartheta \;{\stackrel {\mathrm {(1)} }{=}}-\rho q\sin \vartheta }$

If equation (3) is multiplied with and equation (4) with and added, the result is: ${\displaystyle \cos \vartheta }$ ${\displaystyle -\sin \vartheta }$

${\displaystyle \rho \vartheta '{\big (}\cos ^{2}\vartheta +\sin ^{2}\vartheta {\big )}={\frac {\rho }{p}}\cos ^{2}\vartheta +\rho q\sin ^{2}\vartheta >0}$, or.

(5) , ${\displaystyle \quad \vartheta '={\frac {1}{p}}\cos ^{2}\vartheta +q\sin ^{2}\vartheta >0}$

${\displaystyle \vartheta }$ is therefore monotonically growing.

It remains to show that is unlimited. ${\displaystyle \vartheta }$

If it were limited, the limit values would exist and and it would be . In particular, is or . ${\displaystyle \vartheta }$ ${\displaystyle \alpha :=\lim _{x\to \infty }\cos ^{2}\vartheta (x)}$ ${\displaystyle \beta :=\lim _{x\to \infty }\sin ^{2}\vartheta (x)}$ ${\displaystyle \alpha +\beta =1}$ ${\displaystyle \alpha >0}$ ${\displaystyle \beta >0}$

In the following, be so big that for everyone . Then equation (5) yields for all after integration ${\displaystyle x_{0}>a}$ ${\displaystyle \cos ^{2}\vartheta (x)\geq {\tfrac {\alpha }{2}},\;\sin ^{2}\vartheta (x)\geq {\tfrac {\beta }{2}}}$ ${\displaystyle x>x_{0}}$ ${\displaystyle x>x_{0}}$

${\displaystyle {\begin{aligned}\vartheta (x)-\vartheta (x_{0})&=\int _{x_{0}}^{x}\vartheta '(t)\mathrm {d} t\\&=\int _{x_{0}}^{x}{\bigg (}{\frac {1}{p(t)}}\underbrace {\cos ^{2}\vartheta (x)} _{\geq \alpha /2}+q(t)\underbrace {\sin ^{2}\vartheta (x)} _{\geq \beta /2}{\bigg )}\mathrm {d} t\\&\geq {\frac {\alpha }{2}}\int _{x_{0}}^{x}{\frac {1}{p(t)}}\mathrm {d} t+{\frac {\beta }{2}}\int _{x_{0}}^{x}q(t)\mathrm {d} t\quad {\xrightarrow[{x\to \infty }]{\quad }}\infty \end{aligned}}}$

a contradiction to the requirement. is therefore unlimited. ${\displaystyle \vartheta }$

${\displaystyle \Box }$

Orthogonal relation

If the Sturm-Liouville operator satisfies the Sturm-Louiville differential equation with a suitable and eigenfunction , then the eigenfunctions form an orthogonal basis in the Hilbert space of square-integrable functions. Hence for ${\displaystyle \textstyle {\mathcal {L}}={\frac {1}{w}}\left(-{\frac {\mathrm {d} }{\mathrm {d} x}}\,p\,{\frac {\mathrm {d} }{\mathrm {d} x}}+q\right)}$ ${\displaystyle w,p,q}$ ${\displaystyle \psi _{n}}$ ${\displaystyle {\mathcal {L}}\psi _{n}=\lambda \psi _{n}}$ ${\displaystyle \psi _{n}}$ ${\displaystyle L^{2}([a,b],w(x)\mathrm {d} x)}$ ${\displaystyle \psi _{n}\neq \psi _{m}}$

${\displaystyle \langle \psi _{n},\psi _{m}\rangle =\int _{a}^{b}\psi _{n}\psi _{m}w\mathrm {d} x=0.}$

proof

With the Sturm-Liouville operator the following equations result for the eigenfunctions : ${\displaystyle {\mathcal {L}}={\frac {1}{w}}\left(-{\frac {\mathrm {d} }{\mathrm {d} x}}\,p\,{\frac {\mathrm {d} }{\mathrm {d} x}}+q\right)}$ ${\displaystyle \psi _{n},\psi _{m}}$

(1) ${\displaystyle \quad \psi _{m}{\mathcal {L}}\psi _{n}=\psi _{m}{\frac {1}{w}}\left(-{\frac {\mathrm {d} }{\mathrm {d} x}}\,p\,{\frac {\mathrm {d} }{\mathrm {d} x}}+q\right)\psi _{n}=\lambda _{n}\psi _{m}\psi _{n}}$

and

(2) ${\displaystyle \quad \psi _{n}{\mathcal {L}}\psi _{m}=\psi _{n}{\frac {1}{w}}\left(-{\frac {\mathrm {d} }{\mathrm {d} x}}\,p\,{\frac {\mathrm {d} }{\mathrm {d} x}}+q\right)\psi _{m}=\lambda _{m}\psi _{n}\psi _{m}}$

If equation (1) is subtracted from equation (2) , the two equations result:

(3) ${\displaystyle \quad \psi _{n}{\mathcal {L}}\psi _{m}-\psi _{m}{\mathcal {L}}\psi _{n}=\psi _{n}{\frac {1}{w}}\left(-{\frac {\mathrm {d} }{\mathrm {d} x}}\,p\,{\frac {\mathrm {d} }{\mathrm {d} x}}+q\right)\psi _{m}-\psi _{m}{\frac {1}{w}}\left(-{\frac {\mathrm {d} }{\mathrm {d} x}}\,p\,{\frac {\mathrm {d} }{\mathrm {d} x}}+q\right)\psi _{n}}$

and

(4) . ${\displaystyle \quad \psi _{n}{\mathcal {L}}\psi _{m}-\psi _{m}{\mathcal {L}}\psi _{n}=(\lambda _{m}-\lambda _{n})\psi _{m}\psi _{n}}$

Using the Lagrange identity for boundary value problems , equation (3) can be summarized as:

(5) ${\displaystyle \quad {\begin{aligned}\psi _{n}{\mathcal {L}}\psi _{m}-\psi _{m}{\mathcal {L}}\psi _{n}&={\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}p(\psi _{n}\psi _{m}'-\psi _{m}\psi _{n}'){\bigg )}\\&={\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}pW(\psi _{n},\psi _{m}){\bigg )}\end{aligned}}}$

where means the Wronsky determinant of the functions . ${\displaystyle W(\psi _{n},\psi _{m})}$ ${\displaystyle \psi _{n},\psi _{m}}$

To calculate the Wronsky determinant by means of the Abelian identity , the differential equation is considered in the illustration , with and . The coefficient matrix of the fundamental system is then and its trace is . Thus the Abelian identity is: ${\displaystyle \textstyle {\mathcal {L}}\psi =\left(-{\frac {\mathrm {d} }{\mathrm {d} x}}\,p\,{\frac {\mathrm {d} }{\mathrm {d} x}}+q\right)\psi =-p\psi ''-p'\psi '+q\psi =0}$ ${\displaystyle \psi ''-a_{1}\psi '-a_{0}\psi =0}$ ${\displaystyle a_{0}={\tfrac {q}{p}}\in C^{1}((a,b),\mathbb {R} )}$ ${\displaystyle a_{1}=-{\tfrac {p'}{p}}\in C^{1}((a,b),\mathbb {R} )}$ ${\displaystyle \left({\begin{smallmatrix}0&1\\{\tfrac {q}{p}}&-{\tfrac {p'}{p}}\end{smallmatrix}}\right)}$ ${\displaystyle \mathrm {Spur} {\Bigg (}\left({\begin{smallmatrix}0&1\\{\tfrac {q}{p}}&-{\tfrac {p'}{p}}\end{smallmatrix}}\right){\Bigg )}=-{\frac {p'}{p}}}$

${\displaystyle W(\psi _{n},\psi _{m})(x)=W(\psi _{n},\psi _{m})(a)\ \exp \left(-\int _{a}^{x}{\frac {p'(\xi )}{p(\xi )}}\,\mathrm {d} \xi \right)}$.

Let it be monotonically increasing and therefore the integral can be represented by and accordingly ${\displaystyle p>0}$ ${\displaystyle p'>0}$ ${\displaystyle \textstyle -\int _{a}^{x}{\frac {p'(\xi )}{p(\xi )}}\,\mathrm {d} \xi =-{\big [}\ln |p(\xi )|{\big ]}_{a}^{x}}$

${\displaystyle W(\phi ,\psi )(x)=W(\phi ,\psi )(a)\ \exp \left(-{\bigg [}\ln {\big |}p(\xi ){\big |}{\bigg ]}_{a}^{x}+{\widetilde {C}}\right)}$.

The choice of the integration constants to results in ${\displaystyle {\widetilde {C}}=-\ln p(a)}$

${\displaystyle W(\phi ,\psi )(x)=W(\phi ,\psi )(a)\exp \left(-\ln {\big |}p(x){\big |}\right)=W(\phi ,\psi )(a){\frac {1}{p(x)}}}$

and equation (5) takes the following form:

${\displaystyle {\begin{aligned}\psi _{n}{\mathcal {L}}\psi _{m}-\psi _{m}{\mathcal {L}}\psi _{n}&={\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}pW(\psi _{n},\psi _{m})(a){\frac {1}{p}}{\bigg )}\\&={\frac {-1}{w}}{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}W(\psi _{n},\psi _{m})(a){\bigg )}\end{aligned}}}$

After transforming and separating the variables , the equation is now:

${\displaystyle -w{\Big (}\psi _{n}{\mathcal {L}}\psi _{m}-\psi _{m}{\mathcal {L}}\psi _{n}{\Big )}\mathrm {d} x=\mathrm {d} {\Big (}W(\psi _{n},\psi _{m})(a){\Big )}}$.

On both sides of the equation there are now one-dimensional Pfaffian forms and since is a constant function, the following applies . A suitable parameterization must be selected for the calculation of the remaining Pfaffian form . The integral is now: ${\displaystyle W(\psi _{n},\psi _{m})(a)}$ ${\displaystyle \mathrm {d} {\Big (}W(\psi _{n},\psi _{m})(a){\Big )}=0}$ ${\displaystyle \varphi (t)=t,\varphi (t_{0})=a,\varphi (t_{1})=b,{\dot {\varphi }}(t)=1}$

${\displaystyle \int _{\varphi }\omega =\int _{a}^{b}\omega _{\varphi (t)}({\dot {\varphi }}(t))\,\mathrm {d} t=\int _{a}^{b}w{\Big (}\psi _{n}{\mathcal {L}}\psi _{m}-\psi _{m}{\mathcal {L}}\psi _{n}{\Big )}\mathrm {d} t=0}$.

Accordingly, the integral vanishes along the interval , so that using equation (4) : ${\displaystyle [a,b]}$

${\displaystyle 0=(\lambda _{m}-\lambda _{n})\int _{a}^{b}w\psi _{m}\psi _{n}\mathrm {d} t}$

However, this condition can only be met if:

${\displaystyle \langle \psi _{n},\psi _{m}\rangle =\langle \psi _{m},\psi _{n}\rangle =0}$.

${\displaystyle \Box }$

Comparison set

Sturm's comparative theorem provides a relationship between the two differential equations

(1) ${\displaystyle \quad {\mathcal {L}}_{1}\phi =-{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}p(x){\frac {\mathrm {d} }{\mathrm {d} x}}\phi (x){\bigg )}+q_{1}(x)\phi (x)=0\qquad }$

(2) , ${\displaystyle \quad {\mathcal {L}}_{2}\psi =-{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}p(x){\frac {\mathrm {d} }{\mathrm {d} x}}\psi (x){\bigg )}+q_{2}(x)\psi (x)=0}$

where is assumed for ${\displaystyle x\in (c,d)\subseteq (a,b)}$

${\displaystyle p(x)>0}$ growing monotonously

${\displaystyle q_{1}(x)\geq q_{2}(x)>0}$ growing monotonously.

If there is a non-trivial solution of the differential equation and a non-trivial solution of , then there is a zero of in the interval between two zeros of . ${\displaystyle \phi }$ ${\displaystyle {\mathcal {L}}_{1}\phi }$ ${\displaystyle \psi }$ ${\displaystyle {\mathcal {L}}_{2}\psi }$ ${\displaystyle (c,d)}$ ${\displaystyle \phi }$ ${\displaystyle \psi }$

proof

The Lagrange identity for boundary value problems is considered as the starting point for the following proof . By equation (1) from the left with multiplied by equation (2) , which is also from the left and subtracted so obtained is multiplied, and a Lagrange identity: ${\displaystyle \psi }$ ${\displaystyle \phi }$

${\displaystyle {\begin{aligned}\phi {\mathcal {L}}_{2}\psi -\psi {\mathcal {L}}_{1}\phi &=\phi {\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}-p\,{\frac {\mathrm {d} }{\mathrm {d} x}}\psi {\bigg )}+\phi q_{2}\psi -\psi {\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}-p\,{\frac {\mathrm {d} }{\mathrm {d} x}}\phi {\bigg )}-\psi q_{1}\phi \\&=\phi {\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}-p\,{\frac {\mathrm {d} }{\mathrm {d} x}}\psi {\bigg )}-\psi {\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}-p\,{\frac {\mathrm {d} }{\mathrm {d} x}}\phi {\bigg )}+{\big (}q_{2}-q_{1}{\big )}\phi \psi \\&=-{\frac {\mathrm {d} }{\mathrm {d} x}}{\bigg (}pW(\phi ,\psi ){\bigg )}+{\big (}q_{2}-q_{1}{\big )}\phi \psi ,\end{aligned}}}$

where indicates the Wronsky determinant of the functions . If the Paff's forms are now formed for this equation , with a suitable parameterization being given by and consequently the variable being replaced by the parameter , the differential equation assumes the following integral representation: ${\displaystyle W(\phi ,\psi )=\left|{\begin{smallmatrix}\phi &\psi \\\phi '&\psi '\end{smallmatrix}}\right|}$ ${\displaystyle \phi ,\psi }$ ${\displaystyle \varphi (t)=t,\varphi (t_{0})=a,\varphi (t_{1})=b,{\dot {\varphi }}(t)=1}$ ${\displaystyle x}$ ${\displaystyle t}$

${\displaystyle \underbrace {\int _{c}^{d}{\Big \langle }\phi {\mathcal {L}}_{2}\psi -\psi {\mathcal {L}}_{1}\phi ,\,{\dot {\varphi }}(t){\Big \rangle }\mathrm {d} t} _{\text{Teil 1}}=\underbrace {-\int _{c}^{d}{\Big \langle }{\frac {\mathrm {d} }{\mathrm {d} t}}{\Big (}pW(\phi ,\psi ){\Big )},\,{\dot {\varphi }}(t){\Big \rangle }\mathrm {d} t} _{\text{Teil 2}}+\underbrace {\int _{c}^{d}{\Big \langle }{\big (}q_{2}-q_{1}{\big )}\phi \psi ,\,{\dot {\varphi }}(t){\Big \rangle }\mathrm {d} t} _{\text{Teil 3}}}$.

Part 1

Since according to the amplitude theorem are bounded and are linear operators, it must apply ${\displaystyle \phi ,\psi }$ ${\displaystyle {\mathcal {L_{1}}},{\mathcal {L_{2}}}}$

${\displaystyle \int _{c}^{d}{\Big \langle }\phi (t){\mathcal {L}}_{2}\psi (t)-\psi (t){\mathcal {L}}_{1}\phi (t),\,{\dot {\varphi }}(t){\Big \rangle }\mathrm {d} t={\widetilde {C}}}$.

Part 2

With the Abelian identity, as shown in the section on orthogonal relation, the following relationship arises:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\Big (}p(t)W(\phi ,\psi )(t){\Big )}={\frac {\mathrm {d} }{\mathrm {d} t}}{\Big (}p(t)W(\phi ,\psi )(c){\frac {1}{p(t)}}{\Big )}={\frac {\mathrm {d} }{\mathrm {d} t}}{\Big (}W(\phi ,\psi )(c){\Big )}=0}$. So the integral is now:

${\displaystyle -\int _{c}^{d}{\Big \langle }{\frac {\mathrm {d} }{\mathrm {d} t}}{\Big (}p(t)W(\phi ,\psi )(t){\Big )},\,{\dot {\varphi }}(t){\Big \rangle }\mathrm {d} t=-\int _{c}^{d}{\Big \langle }0,\,{\dot {\varphi }}(t){\Big \rangle }\mathrm {d} t=0}$

part 3

Since the functions satisfy the amplitude theorem and are monotonically decreasing, the integral remains restricted in the interval and the following applies: ${\displaystyle \phi ,\psi }$ ${\displaystyle q_{2}-q_{1}<0}$ ${\displaystyle (c,d)}$

${\displaystyle \int _{c}^{d}{\Big \langle }{\big (}q_{2}(t)-q_{1}(t){\big )}\phi (t)\psi (t),\,{\dot {\varphi }}(t){\Big \rangle }\mathrm {d} t={\widetilde {\widetilde {C}}}}$.

With this integral equation it becomes clear that must apply . ${\displaystyle {\widetilde {C}}={\widetilde {\widetilde {C}}}}$

In order statements about the course of the proper functions within the interval to make the following construction is considered: . ${\displaystyle (c,d)}$ ${\displaystyle -{\bigg [}pW(\phi ,\psi ){\bigg ]}_{c}^{d}}$

If the two linearly independent functions and oBdA are given, it follows from equation (2) that and thus the Wronski determinant can be represented as follows ${\displaystyle \psi }$ ${\displaystyle \phi :=p\psi '}$ ${\displaystyle -(p\psi ')'+q_{2}\psi =0}$ ${\displaystyle \phi '=(p\psi ')'=q_{2}\psi }$

${\displaystyle W(\phi ,\psi )={\begin{vmatrix}\phi &\psi \\\phi '&\psi '\end{vmatrix}}={\begin{vmatrix}p\psi '&\psi \\q_{2}\psi &\psi '\end{vmatrix}}=p\left(\psi '\right)^{2}-q_{2}\left(\psi \right)^{2}}$

and therefore

${\displaystyle -{\bigg [}pW(\phi ,\psi ){\bigg ]}_{c}^{d}=-{\bigg [}p\left(p\left(\psi '\right)^{2}-q_{2}\left(\psi \right)^{2}\right){\bigg ]}_{c}^{d}}$.

Now be o. B. d. A. on the interval such that the Dirichlet boundary condition is fulfilled, then follows ${\displaystyle \phi =p\psi '\geq 0}$ ${\displaystyle (c,d)\subseteq (a,b)}$ ${\displaystyle \phi (c)=p\psi '(c)=0=p\psi '(d)=\phi (d)}$

${\displaystyle {\begin{aligned}-{\bigg [}pW(\phi ,\psi ){\bigg ]}_{c}^{d}&=-{\bigg [}p\left(p\left(\psi '\right)^{2}-q_{2}\left(\psi \right)^{2}\right){\bigg ]}_{c}^{d}={\bigg [}pq_{2}\left(\psi \right)^{2}{\bigg ]}_{c}^{d}\\&=p(d)q_{2}(d){\big (}\psi (d){\big )}^{2}-p(c)q_{2}(c){\big (}\psi (c){\big )}^{2}.\end{aligned}}}$

To show which sign it has, the law of amplitude is used and the following inequalities are considered with the identity ${\displaystyle \left(p(d)q_{2}(d)\left(\psi (d)\right)^{2}-p(c)q_{2}(c)\left(\psi (c)\right)^{2}\right)}$ ${\displaystyle (pq_{2})'>0}$ ${\displaystyle |\psi (d)|<|\psi (c)|}$ ${\displaystyle (\psi )^{2}=|\psi |^{2}}$

(3) and ${\displaystyle \quad p(d)q_{2}(d)\left({\big (}\psi (d){\big )}^{2}-{\big (}\psi (c){\big )}^{2}\right)<0}$

(4) ${\displaystyle \quad p(c)q_{2}(c)\left({\big (}\psi (d){\big )}^{2}-{\big (}\psi (c){\big )}^{2}\right)<0.}$

Adding (3) and (4) yields

${\displaystyle p(d)q_{2}(d)\left({\big (}\psi (d){\big )}^{2}-{\big (}\psi (c){\big )}^{2}\right)+p(c)q_{2}(c)\left({\big (}\psi (d){\big )}^{2}-{\big (}\psi (c){\big )}^{2}\right)<0}$.

After re-sorting it becomes

${\displaystyle p(d)q_{2}(d)\left(\psi (d)\right)^{2}-p(c)q_{2}(c)\left(\psi (c)\right)^{2}-p(c)q_{2}(c)\left(\psi (d)\right)^{2}-p(d)q_{2}(d)\left(\psi (c)\right)^{2}<0}$.

According to the prerequisite , and thus or and consequently must apply ${\displaystyle p(c)q_{2}(c)<p(d)q_{2}(d)}$ ${\displaystyle \left(\psi (d)\right)^{2}<\left(\psi (c)\right)^{2}}$ ${\displaystyle p(c)q_{2}(c)\left(\psi (d)\right)^{2}<p(d)q_{2}(d)\left(\psi (c)\right)^{2}}$ ${\displaystyle p(c)q_{2}(c)\left(\psi (d)\right)^{2}-p(d)q_{2}(d)\left(\psi (c)\right)^{2}<0}$

${\displaystyle p(d)q_{2}(d)\left(\psi (d)\right)^{2}-p(c)q_{2}(c)\left(\psi (c)\right)^{2}<0}$.

So it applies

${\displaystyle -{\bigg [}pW(\phi ,\psi ){\bigg ]}_{c}^{d}<0}$.

Because of the Dirichlet boundary condition, and it holds . Since by assumption on , there is according to the intermediate value set a so that occupies a local extreme point. Below this extreme point is monotonically increasing and above the extreme point is monotonously decreasing. Correspondingly, in is initially monotonically increasing and then monotonically decreasing and because of the change in sign from in must have a zero in . ${\displaystyle \phi (c)=p\psi '(c)=0=p\psi '(d)=\phi (d)}$ ${\displaystyle \psi '(c)=0=\psi '(d)}$ ${\displaystyle \phi \geq 0}$ ${\displaystyle (c,d)}$ ${\displaystyle \xi \in (c,d)}$ ${\displaystyle \phi }$ ${\displaystyle \phi }$ ${\displaystyle \phi }$ ${\displaystyle \psi '}$ ${\displaystyle (c,d)}$ ${\displaystyle \psi '}$ ${\displaystyle (c,d)}$ ${\displaystyle \psi }$ ${\displaystyle (c,d)}$

${\displaystyle \Box }$

example

A simple example is the differential equation

${\displaystyle -\psi ''=\lambda \psi }$

on the interval , together with the Dirichlet boundary conditions ${\displaystyle [0,\pi ]}$

${\displaystyle \psi (0)=\psi (\pi )=0.}$

Due to the boundary conditions, the periodic approach is chosen for and arbitrary . Because is and so and therefore for . The sequence of the eigenvalues ​​is therefore ${\displaystyle \psi (x)=a\sin({\sqrt {\lambda }}x)}$ ${\displaystyle \lambda >0}$ ${\displaystyle a\in \mathbb {R} }$ ${\displaystyle \psi (0)=\psi (\pi )=0}$ ${\displaystyle a\neq 0}$ ${\displaystyle \sin({\sqrt {\lambda }}\pi )=0,}$ ${\displaystyle {\sqrt {\lambda }}\pi =n\pi }$ ${\displaystyle \lambda =n^{2}}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle \lambda _{n}=n^{2}}$

and satisfies the Weyl asymptotics. The sequence of eigenfunctions results, except for the coefficients to be determined to ${\displaystyle a_{n}}$

${\displaystyle \psi _{n}(x)=a_{n}\sin(n\,x).}$

The orthonormal basis of eigenfunctions in Hilbert space with results using the trigonometric formula : ${\displaystyle L^{2}([a,b],\mathrm {d} x)}$ ${\displaystyle w(x)=1}$ ${\displaystyle \textstyle \sin(nx)\;\sin(mx)={\frac {1}{2}}{\Big (}\cos {\big (}(n-m)x{\big )}-\cos {\big (}(n+m)x{\big )}{\Big )}}$

${\displaystyle {\begin{aligned}\langle \psi _{n},\psi _{m}\rangle &=\int {\overline {\psi _{n}(x)}}\psi _{m}(x)\mathrm {d} x=\int _{0}^{\pi }{\overline {a_{n}\sin(nx)}}\;a_{m}\sin(mx)\,\mathrm {d} x=a_{n}a_{m}\int _{0}^{\pi }\sin(nx)\sin(mx)\,\mathrm {d} x\\&={\frac {a_{n}a_{m}}{2}}\int _{0}^{\pi }{\bigg (}\cos {\big (}(n-m)x{\big )}-\cos {\big (}(n+m)x{\big )}{\bigg )}\,\mathrm {d} x\\\\&={\begin{cases}{\frac {a_{n}a_{m}}{2}}{\bigg [}{\frac {1}{n-m}}\sin {\big (}(n-m)x{\big )}-{\frac {1}{n+m}}\sin {\big (}(n+m)x{\big )}{\bigg ]}_{0}^{\pi }=0&&{\text{wenn}}\;n\neq m\\\\{\frac {a_{n}^{2}}{2}}{\Bigg [}x-{\frac {1}{2n}}\sin(2nx){\bigg ]}_{0}^{\pi }={\frac {a_{n}^{2}\pi }{2}}&&{\text{wenn}}\;n=m\end{cases}}\\\\&={\frac {a_{n}^{2}\pi }{2}}\delta _{nm}.\end{aligned}}}$

Here, the Kronecker delta and the normalization mean conditional , so that the normalized eigenfunctions represent the representation ${\displaystyle \delta _{nm}}$ ${\displaystyle \langle \psi _{n},\psi _{m}\rangle =\delta _{nm}}$ ${\displaystyle a_{n}={\sqrt {\frac {2}{\pi }}}}$

${\displaystyle \psi _{n}(x)={\sqrt {\frac {2}{\pi }}}\sin(n\,x)}$

accept.

The associated eigenfunction expansion is the Fourier series with

${\displaystyle \Psi =\sum _{n=1}^{\infty }\psi _{n}=\sum _{n=1}^{\infty }{\sqrt {\frac {2}{\pi }}}\sin(nx).}$

Mathematical theory

The suitable mathematical framework is the Hilbert space with the scalar product ${\displaystyle L^{2}([a,b];w(x){\rm {d}}x)}$

${\displaystyle \langle f,g\rangle :=\int _{a}^{b}{\overline {f(x)}}g(x)w(x)\,{\rm {d}}x}$.

In this space there is a self-adjoint operator if it is defined on the set of (in the sense of the weak derivative ) differentiable functions that satisfy the boundary conditions: ${\displaystyle {\mathcal {L}}}$

${\displaystyle {\begin{aligned}{\mathfrak {D}}({\mathcal {L}})=\{&f\in L^{2}([a,b];w(x){\rm {d}}x):f,pf'\in AC[a,b],\,{\mathcal {L}}f\in L^{2}([a,b];w(x){\rm {d}}x),\\&,\,\cos(\alpha )f(a)+\sin(\alpha )p(a)f'(a)=\cos(\beta )f(b)+\sin(\beta )p(b)f'(b)=0\}.\end{aligned}}}$

Here denotes the set of functions that are absolutely continuous . Since is an unbounded operator, consider the resolvent ${\displaystyle AC[a,b]}$ ${\displaystyle [a,b]}$ ${\displaystyle {\mathcal {L}}}$

${\displaystyle ({\mathcal {L}}-z)^{-1},\qquad z\in \mathbb {C} }$,

where there must be no eigenvalue. It turns out that the resolvent is an integral operator with a continuous kernel ( Green's function of the boundary value problem). Thus the resolvent is a compact operator , and the existence of a countable sequence of eigenfunctions follows from the spectral theorem for compact operators. ${\displaystyle z}$

The relationship between the eigenvalues ​​of and the resolvent follows, since it is equivalent to with . ${\displaystyle {\mathcal {L}}}$ ${\displaystyle ({\mathcal {L}}-z)^{-1}\psi =\alpha \psi }$ ${\displaystyle {\mathcal {L}}\psi =\lambda \psi }$ ${\displaystyle \lambda =(z+\alpha ^{-1})}$

Singular Sturm Liouville Problems

If the above conditions are not met, one speaks of a singular Sturm-Liouville problem . The spectrum then generally no longer consists only of eigenvalues ​​and also has a continuous component. There are also generalized eigenfunctions, and the associated eigenfunction expansion is an integral transformation (compare Fourier transformation instead of Fourier series ).

Change or the sign on the interval , one speaks of an indefinite Sturm-Liouville problem . ${\displaystyle p}$ ${\displaystyle w}$ ${\displaystyle [a,b]}$

application

• The case corresponds to the one-dimensional time-independent Schrödinger equation . ${\displaystyle p(x)=w(x)=1}$
• The separation approach for the solution of partial differential equations leads to Sturm-Liouville problems.

Web links

• Walter Oevel: Storm Liouville Problems. (PDF; 314 kB)

literature

• Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (=  Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( univie.ac.at ).
• Wolfgang Walter : Ordinary differential equations . 7th edition. Springer, Berlin 2000, ISBN 3-540-67642-2 .
• Harro Heuser : Ordinary differential equations . Vieweg + Teubner, Wiesbaden 2009 (6th edition), ISBN 978-3-8348-0705-2
• Joachim Weidmann : Linear operators in Hilbert spaces . Part 2. Applications. Teubner, Stuttgart / Leipzig / Wiesbaden 2003, ISBN 3-519-02237-0 .

References and comments

1. ^ Charles-François Sturm : Sur le développement des fonctions ou parties de fonctions en séries dont les divers terms sont assujettis à satisfaire à une même équation différentielle du second ordre contenant un paramètre variable, Journal de mathématiques, 1836, bibnum
2. ↑ Harro Heuser : Ordinary differential equations , Vieweg + Teubner 2009 (6th edition), pages 328–338, ISBN 978-3-8348-0705-2
3. ^ Wolfgang Walter : Ordinary differential equations , Springer-Verlag 2000, page 287-290, ISBN 3-540-67642-2