diff options
Diffstat (limited to 'source/know/concept/sturm-liouville-theory/index.md')
| -rw-r--r-- | source/know/concept/sturm-liouville-theory/index.md | 118 |
1 files changed, 59 insertions, 59 deletions
diff --git a/source/know/concept/sturm-liouville-theory/index.md b/source/know/concept/sturm-liouville-theory/index.md index 23922d5..75daae3 100644 --- a/source/know/concept/sturm-liouville-theory/index.md +++ b/source/know/concept/sturm-liouville-theory/index.md @@ -21,23 +21,23 @@ of eigenfunctions. ## General operator Consider the most general form of a second-order linear -differential operator $\hat{L}$, where $p_0(x)$, $p_1(x)$, and $p_2(x)$ -are real functions of $x \in [a,b]$ which are non-zero for all $x \in ]a, b[$: +differential operator $$\hat{L}$$, where $$p_0(x)$$, $$p_1(x)$$, and $$p_2(x)$$ +are real functions of $$x \in [a,b]$$ which are non-zero for all $$x \in ]a, b[$$: $$\begin{aligned} \hat{L} \{u(x)\} = p_0(x) u''(x) + p_1(x) u'(x) + p_2(x) u(x) \end{aligned}$$ We now define the **adjoint** or **Hermitian** operator -$\hat{L}^\dagger$ analogously to matrices: +$$\hat{L}^\dagger$$ analogously to matrices: $$\begin{aligned} \inprod{f}{\hat{L} g} = \inprod{\hat{L}^\dagger f}{g} \end{aligned}$$ -What is $\hat{L}^\dagger$, given the above definition of $\hat{L}$? -We start from the inner product $\inprod{f}{\hat{L} g}$: +What is $$\hat{L}^\dagger$$, given the above definition of $$\hat{L}$$? +We start from the inner product $$\inprod{f}{\hat{L} g}$$: $$\begin{aligned} \inprod{f}{\hat{L} g} @@ -51,7 +51,7 @@ $$\begin{aligned} &= \big[ f^* \big( p_0 g' + (p_1 - p_0') g \big) - (f^*)' p_0 g \big]_a^b + \int_a^b \big( \hat{L}^\dagger\{f\} \big)^* g \dd{x} \end{aligned}$$ -We now have an expression for $\hat{L}^\dagger$, but are left with an +We now have an expression for $$\hat{L}^\dagger$$, but are left with an annoying boundary term: $$\begin{aligned} @@ -60,8 +60,8 @@ $$\begin{aligned} \end{aligned}$$ To fix this, -let us demand that $p_1(x) = p_0'(x)$ and that -$[p_0(f^* g' - (f^*)' g)]_a^b = 0$, leaving: +let us demand that $$p_1(x) = p_0'(x)$$ and that +$$[p_0(f^* g' - (f^*)' g)]_a^b = 0$$, leaving: $$\begin{aligned} \inprod{f}{\hat{L} g} @@ -69,8 +69,8 @@ $$\begin{aligned} = \inprod{\hat{L}^\dagger f}{g} \end{aligned}$$ -Using the aforementioned restriction $p_1(x) = p_0'(x)$, -we then take a look at the definition of $\hat{L}^\dagger$: +Using the aforementioned restriction $$p_1(x) = p_0'(x)$$, +we then take a look at the definition of $$\hat{L}^\dagger$$: $$\begin{aligned} \hat{L}^\dagger \{f\} @@ -83,7 +83,7 @@ $$\begin{aligned} &= (p_0 f')' + p_2 f \end{aligned}$$ -The original operator $\hat{L}$ reduces to the same form, +The original operator $$\hat{L}$$ reduces to the same form, so it is **self-adjoint**: $$\begin{aligned} @@ -93,19 +93,19 @@ $$\begin{aligned} = \hat{L}^\dagger \{f\} \end{aligned}$$ -Consequently, every such second-order linear operator $\hat{L}$ is self-adjoint, -as long as it satisfies the constraints $p_1(x) = p_0'(x)$ and $[p_0 (f^* g' - (f^*)' g)]_a^b = 0$. +Consequently, every such second-order linear operator $$\hat{L}$$ is self-adjoint, +as long as it satisfies the constraints $$p_1(x) = p_0'(x)$$ and $$[p_0 (f^* g' - (f^*)' g)]_a^b = 0$$. Let us ignore the latter constraint for now (it will return later), -and focus on the former: what if $\hat{L}$ does not satisfy $p_0' \neq p_1$? -We multiply it by an unknown $p(x) \neq 0$, and divide by $p_0(x) \neq 0$: +and focus on the former: what if $$\hat{L}$$ does not satisfy $$p_0' \neq p_1$$? +We multiply it by an unknown $$p(x) \neq 0$$, and divide by $$p_0(x) \neq 0$$: $$\begin{aligned} \frac{p(x)}{p_0(x)} \hat{L} \{u\} = p(x) u'' + p(x) \frac{p_1(x)}{p_0(x)} u' + p(x) \frac{p_2(x)}{p_0(x)} u \end{aligned}$$ -We now define $q(x)$, -and demand that the derivative $p'(x)$ of the unknown $p(x)$ satisfies: +We now define $$q(x)$$, +and demand that the derivative $$p'(x)$$ of the unknown $$p(x)$$ satisfies: $$\begin{aligned} q(x) = p(x) \frac{p_2(x)}{p_0(x)} @@ -113,7 +113,7 @@ $$\begin{aligned} p'(x) = p(x) \frac{p_1(x)}{p_0(x)} \end{aligned}$$ -The latter is a differential equation for $p(x)$, which we solve by integration: +The latter is a differential equation for $$p(x)$$, which we solve by integration: $$\begin{gathered} \frac{p_1(x)}{p_0(x)} = \frac{1}{p(x)} \dv{p}{x} @@ -128,7 +128,7 @@ $$\begin{gathered} p(x) = p(a) \exp\!\Big( \int_a^x \frac{p_1(\xi)}{p_0(\xi)} \dd{\xi} \Big) \end{gathered}$$ -Now that we have $p(x)$ and $q(x)$, we can define a new operator $\hat{L}_p$ as follows: +Now that we have $$p(x)$$ and $$q(x)$$, we can define a new operator $$\hat{L}_p$$ as follows: $$\begin{aligned} \hat{L}_p \{u\} @@ -138,11 +138,11 @@ $$\begin{aligned} \end{aligned}$$ This is the self-adjoint form from earlier! -So even if $p_0' \neq p_1$, any second-order linear operator with $p_0(x) \neq 0$ +So even if $$p_0' \neq p_1$$, any second-order linear operator with $$p_0(x) \neq 0$$ can easily be put in self-adjoint form. -This general form is known as the **Sturm-Liouville operator** $\hat{L}_{SL}$, -where $p(x)$ and $q(x)$ are non-zero real functions of the variable $x \in [a,b]$: +This general form is known as the **Sturm-Liouville operator** $$\hat{L}_{SL}$$, +where $$p(x)$$ and $$q(x)$$ are non-zero real functions of the variable $$x \in [a,b]$$: $$\begin{aligned} \boxed{ @@ -156,8 +156,8 @@ $$\begin{aligned} ## Eigenvalue problem A **Sturm-Liouville problem** (SLP) is analogous to a matrix eigenvalue problem, -where $w(x)$ is a real weight function, $\lambda$ is the **eigenvalue**, -and $u(x)$ is the corresponding **eigenfunction**: +where $$w(x)$$ is a real weight function, $$\lambda$$ is the **eigenvalue**, +and $$u(x)$$ is the corresponding **eigenfunction**: $$\begin{aligned} \boxed{ @@ -165,14 +165,14 @@ $$\begin{aligned} } \end{aligned}$$ -Necessarily, $w(x) > 0$ except in isolated points, where $w(x) = 0$ is allowed; -the point is that any inner product $\Inprod{f}{w g}$ may never be zero due to $w$'s fault. -Furthermore, the convention is that $u(x)$ cannot be trivially zero. +Necessarily, $$w(x) > 0$$ except in isolated points, where $$w(x) = 0$$ is allowed; +the point is that any inner product $$\Inprod{f}{w g}$$ may never be zero due to $$w$$'s fault. +Furthermore, the convention is that $$u(x)$$ cannot be trivially zero. -In our derivation of $\hat{L}_{SL}$, +In our derivation of $$\hat{L}_{SL}$$, we removed a boundary term to get self-adjointness. Consequently, to have a valid SLP, the boundary conditions for -$u(x)$ must be as follows, otherwise the operator cannot be self-adjoint: +$$u(x)$$ must be as follows, otherwise the operator cannot be self-adjoint: $$\begin{aligned} \Big[ p(x) \big( u^*(x) u'(x) - (u'(x))^* u(x) \big) \Big]_a^b = 0 @@ -181,16 +181,16 @@ $$\begin{aligned} There are many boundary conditions (BCs) which satisfy this requirement. Some notable ones are listed here non-exhaustively: -+ **Dirichlet BCs**: $u(a) = u(b) = 0$ -+ **Neumann BCs**: $u'(a) = u'(b) = 0$ -+ **Robin BCs**: $\alpha_1 u(a) + \beta_1 u'(a) = \alpha_2 u(b) + \beta_2 u'(b) = 0$ with $\alpha_{1,2}, \beta_{1,2} \in \mathbb{R}$ -+ **Periodic BCs**: $p(a) = p(b)$, $u(a) = u(b)$, and $u'(a) = u'(b)$ -+ **Legendre "BCs"**: $p(a) = p(b) = 0$ ++ **Dirichlet BCs**: $$u(a) = u(b) = 0$$ ++ **Neumann BCs**: $$u'(a) = u'(b) = 0$$ ++ **Robin BCs**: $$\alpha_1 u(a) + \beta_1 u'(a) = \alpha_2 u(b) + \beta_2 u'(b) = 0$$ with $$\alpha_{1,2}, \beta_{1,2} \in \mathbb{R}$$ ++ **Periodic BCs**: $$p(a) = p(b)$$, $$u(a) = u(b)$$, and $$u'(a) = u'(b)$$ ++ **Legendre "BCs"**: $$p(a) = p(b) = 0$$ Once this requirement is satisfied, Sturm-Liouville theory gives us -some very useful information about $\lambda$ and $u(x)$. +some very useful information about $$\lambda$$ and $$u(x)$$. From the definition of an SLP, we know that, given two arbitrary (and possibly identical) -eigenfunctions $u_n$ and $u_m$, the following must be satisfied: +eigenfunctions $$u_n$$ and $$u_m$$, the following must be satisfied: $$\begin{aligned} 0 = \hat{L}_{SL}\{u_n\} + \lambda_n w u_n = \hat{L}_{SL}\{u_m^*\} + \lambda_m^* w u_m^* @@ -215,7 +215,7 @@ $$\begin{aligned} &= (\lambda_m^* - \lambda_n) \Inprod{u_m}{w u_n} \end{aligned}$$ -The operator $\hat{L}_{SL}$ is self-adjoint by definition, +The operator $$\hat{L}_{SL}$$ is self-adjoint by definition, so the left-hand side vanishes, leaving us with: $$\begin{aligned} @@ -223,20 +223,20 @@ $$\begin{aligned} &= (\lambda_m^* - \lambda_n) \Inprod{u_m}{w u_n} \end{aligned}$$ -When $m = n$, the inner product $\Inprod{u_n}{w u_n}$ is real and positive -(assuming $u_n$ is not trivially zero, in which case it would be disqualified anyway). -In this case we thus know that $\lambda_n^* = \lambda_n$, -i.e. the eigenvalue $\lambda_n$ is real for any $n$. +When $$m = n$$, the inner product $$\Inprod{u_n}{w u_n}$$ is real and positive +(assuming $$u_n$$ is not trivially zero, in which case it would be disqualified anyway). +In this case we thus know that $$\lambda_n^* = \lambda_n$$, +i.e. the eigenvalue $$\lambda_n$$ is real for any $$n$$. -When $m \neq n$, then $\lambda_m^* - \lambda_n$ may or may not be zero, +When $$m \neq n$$, then $$\lambda_m^* - \lambda_n$$ may or may not be zero, depending on the degeneracy. If there is no degeneracy, we -see that $\Inprod{u_m}{w u_n} = 0$, i.e. the eigenfunctions are orthogonal. +see that $$\Inprod{u_m}{w u_n} = 0$$, i.e. the eigenfunctions are orthogonal. In case of degeneracy, manual orthogonalization is needed, but as it turns out, this is guaranteed to be doable, using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method/). -In conclusion, **a Sturm-Liouville problem has real eigenvalues $\lambda$, -and all the corresponding eigenfunctions $u(x)$ are mutually orthogonal**: +In conclusion, **a Sturm-Liouville problem has real eigenvalues $$\lambda$$, +and all the corresponding eigenfunctions $$u(x)$$ are mutually orthogonal**: $$\begin{aligned} \boxed{ @@ -252,16 +252,16 @@ so it is always worth checking whether you are dealing with an SLP. Another useful fact of SLPs is that they always have an infinite number of discrete eigenvalues. -Furthermore, the eigenvalues always ascend to $+\infty$; -in other words, there always exists a *lowest* eigenvalue $\lambda_0 > -\infty$, +Furthermore, the eigenvalues always ascend to $$+\infty$$; +in other words, there always exists a *lowest* eigenvalue $$\lambda_0 > -\infty$$, known as the **ground state**. ## Completeness -Not only are the eigenfunctions $u_n(x)$ of an SLP orthogonal, they -also form a **complete basis**, meaning that any well-behaved function $f(x)$ can be -expanded as a **generalized Fourier series** with coefficients $a_n$: +Not only are the eigenfunctions $$u_n(x)$$ of an SLP orthogonal, they +also form a **complete basis**, meaning that any well-behaved function $$f(x)$$ can be +expanded as a **generalized Fourier series** with coefficients $$a_n$$: $$\begin{aligned} \boxed{ @@ -271,12 +271,12 @@ $$\begin{aligned} } \end{aligned}$$ -This series will converge significantly faster if $f(x)$ -satisfies the same BCs as $u_n(x)$. In that case the -expansion will even be valid for the inclusive interval $x \in [a, b]$. +This series will converge significantly faster if $$f(x)$$ +satisfies the same BCs as $$u_n(x)$$. In that case the +expansion will even be valid for the inclusive interval $$x \in [a, b]$$. -To find an expression for the coefficients $a_n$, -we multiply the above generalized Fourier series by $w(x) u_m^*(x)$ for an arbitrary $m$: +To find an expression for the coefficients $$a_n$$, +we multiply the above generalized Fourier series by $$w(x) u_m^*(x)$$ for an arbitrary $$m$$: $$\begin{aligned} f(x) w(x) u_m^*(x) @@ -303,9 +303,9 @@ $$\begin{aligned} = a_m A_m \end{aligned}$$ -After isolating this for $a_n$, we see that +After isolating this for $$a_n$$, we see that the coefficients are given by the projection of the target -function $f(x)$ onto the normalized eigenfunctions $u_n(x) / A_n$: +function $$f(x)$$ onto the normalized eigenfunctions $$u_n(x) / A_n$$: $$\begin{aligned} \boxed{ @@ -317,7 +317,7 @@ $$\begin{aligned} As a final remark, we can see something interesting by rearranging the generalized Fourier series -after inserting the expression for $a_n$: +after inserting the expression for $$a_n$$: $$\begin{aligned} f(x) @@ -328,7 +328,7 @@ $$\begin{aligned} \end{aligned}$$ Upon closer inspection, the parenthesized summation -must be the [Dirac delta function](/know/concept/dirac-delta-function/) $\delta(x)$ +must be the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(x)$$ for the integral to work out. This is in fact the underlying requirement for completeness: |