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<h1>Methods Lecture 4</h1>
<p style="text-align: center; color: #666; font-style: italic;">Sturm-Liouville Theory and Abstract Eigenvalue Problems</p>
<!-- ===== CONTENT BOXES ===== -->
<div class="default content-box">
<div class="timestamp" onclick="openVideoSide('00:15', 'Course Announcements', 'default', this)">
<span class="play-icon">▶</span>00:15
</div>
<h3>Course Announcements</h3>
<p>Lecture notes for lectures 1, 2, and 3 are available on Moodle, along with a crib sheet containing the most important facts for the series. The new sections on Sturm-Liouville theory and abstract eigenvalue problems should be included in the crib sheet.</p>
</div>
<div class="default content-box">
<div class="timestamp" onclick="openVideoSide('01:01', 'Review of Finite Dimensional Linear Algebra', 'default', this)">
<span class="play-icon">▶</span>01:01
</div>
<h3>Review of Finite Dimensional Linear Algebra</h3>
<p>Recall from 1A Vectors and Matrices: A linear map $A$ from an $n$-dimensional vector space to itself is called <strong>Hermitian</strong> if $A^\dagger = A$, or equivalently, if $\mathbf{x} \cdot A\mathbf{y} = A\mathbf{x} \cdot \mathbf{y}$ for all $\mathbf{x}, \mathbf{y} \in V_n$.</p>
</div>
<div class="default content-box">
<div class="timestamp" onclick="openVideoSide('02:16', 'Properties of Hermitian Operators', 'default', this)">
<span class="play-icon">▶</span>02:16
</div>
<h3>Properties of Hermitian Operators</h3>
<p>Hermitian operators have several nice properties:</p>
<ol>
<li><strong>Eigenvalues are real</strong></li>
<li><strong>Eigenvectors with distinct eigenvalues are orthogonal</strong></li>
<li><strong>Spectral theorem:</strong> You can pick an orthogonal set of eigenvectors $\{v_i\}$ such that for each $\mathbf{x} \in V_n$, we can write $\mathbf{x} = \sum_{i=1}^N \hat{x}_i v_i$ where $\hat{x}_i = \frac{\mathbf{x} \cdot v_i}{\|v_i\|^2}$</li>
</ol>
</div>
<div class="default content-box">
<div class="timestamp" onclick="openVideoSide('04:49', 'Complex Inner Product Space', 'default', this)">
<span class="play-icon">▶</span>04:49
</div>
<h3>Complex Inner Product Space</h3>
<p>We assume a complex inner product space where the inner product is defined as $\mathbf{x} \cdot \mathbf{y} = \sum_{i=1}^n x_i \overline{y_i}$ (complex conjugated). All discussions of orthogonality refer to this complex inner product.</p>
</div>
<div class="default content-box">
<div class="timestamp" onclick="openVideoSide('05:11', 'Infinite Dimensional Extension', 'default', this)">
<span class="play-icon">▶</span>05:11
</div>
<h3>Infinite Dimensional Extension</h3>
<p>Today's question: <strong>What if $n$ is infinite?</strong> How much of this finite-dimensional theory carries over to infinite-dimensional vector spaces? The infinite is not just "finite but very big" - it's fundamentally different, and unfortunately, some properties from finite-dimensional linear algebra don't carry over.</p>
</div>
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('05:45', 'Function Space Definition', 'definition', this)">
<span class="play-icon">▶</span>05:45
</div>
<h3>Function Space</h3>
<p>We introduce a vector space of "nice" functions $f$ from some closed interval $[a,b]$ to $\mathbb{C}$. The term "nice" will be elaborated on later, but for now, it means functions with sufficient regularity properties.</p>
</div>
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('06:30', 'Weighted Inner Product', 'definition', this)">
<span class="play-icon">▶</span>06:30
</div>
<h3>Weighted Inner Product</h3>
<p>We introduce an inner product on our function space, denoted with subscript $w$:</p>
<p>$$\langle f, g \rangle_w = \int_a^b f(x) \overline{g(x)} w(x) \, dx$$</p>
<p>where $w$ is real-valued and positive on the open interval $(a,b)$. We call $w$ the <strong>weight function</strong> associated with the inner product.</p>
</div>
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('08:02', 'Associated Norm', 'definition', this)">
<span class="play-icon">▶</span>08:02
</div>
<h3>Associated Norm</h3>
<p>With the inner product comes an associated norm:</p>
<p>$$\|f\|_w = \sqrt{\langle f, f \rangle_w}$$</p>
<p>When the weight function is identically 1, we simply write $\langle f, g \rangle$ and $\|f\|$ without the subscript.</p>
</div>
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('09:04', 'Self-Adjoint Differential Operator', 'definition', this)">
<span class="play-icon">▶</span>09:04
</div>
<h3>Self-Adjoint Differential Operator</h3>
<p>A linear differential operator $L$ is said to be <strong>self-adjoint</strong> on $V$ with its inner product if:</p>
<p>$$\langle L y_1, y_2 \rangle_w = \langle y_1, L y_2 \rangle_w$$</p>
<p>for all $y_1, y_2 \in V$.</p>
</div>
<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('10:33', 'Eigenfunction-Eigenvalue Pair', 'definition', this)">
<span class="play-icon">▶</span>10:33
</div>
<h3>Eigenfunction-Eigenvalue Pair</h3>
<p>We say that $(y, \lambda)$ where $y \in V$ (non-zero) and $\lambda \in \mathbb{C}$ is an <strong>eigenfunction-eigenvalue pair</strong> for $L$ if:</p>
<p>$$L y = \lambda y$$</p>
</div>
<div class="proposition content-box">
<div class="timestamp" onclick="openVideoSide('12:00', 'Properties of Self-Adjoint Operators', 'proposition', this)">
<span class="play-icon">▶</span>12:00
</div>
<h3>Properties of Self-Adjoint Operators</h3>
<p>If $L$ is self-adjoint on $V$ with the inner product, then:</p>
<ol>
<li><strong>Eigenvalues are all real</strong></li>
<li><strong>Eigenfunctions with distinct eigenvalues are orthogonal</strong> (with respect to the inner product)</li>
<li><strong>There exists a complete orthogonal set of eigenfunctions</strong> $\{y_n\}_{n=1}^{\infty}$ such that for each $f \in V$, we can write $f = \sum_{n=1}^{\infty} \hat{f}_n y_n$ where $\hat{f}_n = \frac{\langle f, y_n \rangle_w}{\|y_n\|^2}$</li>
</ol>
</div>
<div class="default content-box">
<div class="timestamp" onclick="openVideoSide('14:58', 'Infinite Sum Convergence', 'default', this)">
<span class="play-icon">▶</span>14:58
</div>
<h3>Infinite Sum Convergence</h3>
<p>The infinite sum in the spectral theorem implicitly means the limit of partial sums. The convergence here should be understood as convergence in norm (though for practical purposes in this course, it will likely work out pointwise as well).</p>
</div>
<div class="proof content-box">
<div class="timestamp" onclick="openVideoSide('15:38', 'Proof of Real Eigenvalues', 'proof', this)">
<span class="play-icon">▶</span>15:38
</div>
<h3>Proof of Real Eigenvalues</h3>
<p><strong>Proof of (1):</strong> If $L y = \lambda y$ with $y \neq 0$, then:</p>
<p>$$(\lambda - \overline{\lambda}) \langle y, y \rangle_w = \langle L y, y \rangle_w - \langle y, L y \rangle_w = 0$$</p>
<p>Since $L$ is self-adjoint, the right-hand side is zero. Since $\langle y, y \rangle_w \neq 0$, we have $\lambda = \overline{\lambda}$, so $\lambda$ is real.</p>
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<div class="timestamp" onclick="openVideoSide('17:00', 'Proof of Orthogonality', 'proof', this)">
<span class="play-icon">▶</span>17:00
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<h3>Proof of Orthogonality</h3>
<p><strong>Proof of (2):</strong> If $L y_1 = \lambda_1 y_1$ and $L y_2 = \lambda_2 y_2$ with $\lambda_1 \neq \lambda_2$, then:</p>
<p>$$(\lambda_1 - \lambda_2) \langle y_1, y_2 \rangle_w = \langle L y_1, y_2 \rangle_w - \langle y_1, L y_2 \rangle_w = 0$$</p>
<p>Since $L$ is self-adjoint and $\lambda_1 \neq \lambda_2$, we have $\langle y_1, y_2 \rangle_w = 0$, so the eigenfunctions are orthogonal.</p>
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<div class="non-examinable content-box">
<div class="timestamp" onclick="openVideoSide('18:49', 'Proof by Authority', 'non-examinable', this)">
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<h3>Proof by Authority</h3>
<p><strong>Proof of (3):</strong> This is a proof by authority - David Hilbert said it was true, which implies it was true. This is not admissible in Tripos exams, but it's completely acceptable in Methods. The rigorous proof requires the spectral theorem for compact self-adjoint operators on Hilbert space (covered in Part II Linear Analysis).</p>
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<div class="default content-box">
<div class="timestamp" onclick="openVideoSide('20:15', 'Sturm-Liouville Operators', 'default', this)">
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<h3>Sturm-Liouville Operators and Boundary Values</h3>
<p>We will study eigenvalue problems of the form $L y = \lambda y$ where $L$ is a differential operator, solved inside some interval $x \in [a,b]$ with boundary conditions at $x = a$ and $x = b$. Here $L$ will be self-adjoint and of a particular type: a <strong>Sturm-Liouville operator</strong>.</p>
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<div class="definition content-box">
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<h3>Sturm-Liouville Operator</h3>
<p>A differential operator $L$ is a <strong>Sturm-Liouville operator</strong> on $[a,b]$ if it has the form:</p>
<p>$$L = \frac{1}{w(x)} \left[ -\frac{d}{dx}\left(p(x)\frac{d}{dx}\right) + q(x) \right]$$</p>
<p>where $p$, $q$, and $w$ are real-valued, and $p$ and $w$ are both positive on the open interval $(a,b)$. The function $w$ is called the <strong>weight function</strong>.</p>
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<h3>Eigenvalue Problem Form</h3>
<p>If we multiply $L y = \lambda y$ by $w(x)$, we get:</p>
<p>$$-\frac{d}{dx}\left(p(x)\frac{dy}{dx}\right) + q(x) y = \lambda w(x) y$$</p>
<p>This looks like a differential equation, but we don't know what $\lambda$ is - we're trying to find both the eigenfunctions $y$ and eigenvalues $\lambda$ that satisfy this equation.</p>
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<div class="timestamp" onclick="openVideoSide('28:35', 'Singular Endpoint', 'definition', this)">
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<h3>Singular Endpoint</h3>
<p>For a Sturm-Liouville operator on $[a,b]$, an endpoint $c$ (either $a$ or $b$) is <strong>singular</strong> if $p(c) = 0$, and <strong>non-singular</strong> otherwise.</p>
<p>We will only impose boundary conditions at non-singular endpoints.</p>
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<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('29:56', 'Real Homogeneous Boundary Conditions', 'definition', this)">
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<h3>Real Homogeneous Boundary Conditions</h3>
<p>At each non-singular endpoint $c$, we impose boundary conditions of the form:</p>
<p>$$\alpha_c y(c) + \beta_c y'(c) = 0$$</p>
<p>where $\alpha_c$ and $\beta_c$ are real numbers (not both zero).</p>
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<div class="definition content-box">
<div class="timestamp" onclick="openVideoSide('32:26', 'Function Space V', 'definition', this)">
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<h3>Function Space V</h3>
<p>We work on the vector space $V$ consisting of functions $y$ that are:</p>
<ul>
<li>Twice continuously differentiable on the closed interval $[a,b]$</li>
<li>Bounded on the closed interval with bounded derivatives</li>
<li>Such that $y$ satisfies real homogeneous boundary conditions at each non-singular endpoint</li>
</ul>
<p>We equip $V$ with the inner product $\langle f, g \rangle_w = \int_a^b f(x) \overline{g(x)} w(x) \, dx$.</p>
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<h3>Example 1: Cosine Weight Function</h3>
<p>Consider the problem:</p>
<p>$$-\frac{d}{dx}\left(\cos\left(\frac{x}{2}\right)\frac{dy}{dx}\right) + \sin\left(\frac{x}{2}\right) y = \lambda x y$$</p>
<p>on the interval $[0, \pi]$ with boundary condition $y(0) = 0$.</p>
<p>Here: $p(x) = \cos(x/2)$, $q(x) = \sin(x/2)$, $w(x) = x$.</p>
<p>At $x = \pi$: $p(\pi) = \cos(\pi/2) = 0$, so $x = \pi$ is a singular endpoint. We only need boundary conditions at the non-singular endpoint $x = 0$.</p>
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<div class="timestamp" onclick="openVideoSide('37:48', 'Example 2: Legendre Polynomials', 'example', this)">
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<h3>Example 2: Legendre Polynomials</h3>
<p>Consider the problem:</p>
<p>$$-\frac{d}{dx}\left((1-x^2)\frac{dy}{dx}\right) = \lambda y$$</p>
<p>on the interval $[-1, 1]$.</p>
<p>Here: $p(x) = 1-x^2$, $q(x) = 0$, $w(x) = 1$.</p>
<p>At $x = \pm 1$: $p(\pm 1) = 0$, so both endpoints are singular. No boundary conditions are needed!</p>
<p>The associated inner product is $\langle f, g \rangle = \int_{-1}^1 f(x) \overline{g(x)} \, dx$.</p>
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<div class="timestamp" onclick="openVideoSide('39:34', 'Self-Adjointness of Sturm-Liouville Operators', 'proposition', this)">
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<h3>Self-Adjointness of Sturm-Liouville Operators</h3>
<p>If $L$ is a Sturm-Liouville operator on $[a,b]$ with weight function $w$, and $y_1, y_2$ are twice continuously differentiable functions on the interval, then:</p>
<p>$$\langle L y_1, y_2 \rangle_w - \langle y_1, L y_2 \rangle_w = [p(x) W(y_1, y_2)]_{x=a}^{x=b}$$</p>
<p>where $W(y_1, y_2) = y_1 y_2' - y_2 y_1'$ is the Wronskian.</p>
<p>In particular, if $y_1, y_2 \in V$ (the function space with appropriate boundary conditions), then $L$ is self-adjoint on $V$ with the inner product specified by the weight function.</p>
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<div class="timestamp" onclick="openVideoSide('42:13', 'Proof of Self-Adjointness', 'proof', this)">
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<h3>Proof of Self-Adjointness</h3>
<p>The proof involves expanding the left-hand side using the definition of $L$ and integrating by parts. After cancellation of terms involving $q(x)$, we obtain:</p>
<p>$$\int_a^b \left[ y_1 \frac{d}{dx}(p y_2') - y_2 \frac{d}{dx}(p y_1') \right] dx = [p(x) W(y_1, y_2)]_{x=a}^{x=b}$$</p>
<p>This follows from the fundamental theorem of calculus and the fact that $\frac{d}{dx}[p(x) W(y_1, y_2)] = y_1 \frac{d}{dx}(p y_2') - y_2 \frac{d}{dx}(p y_1')$.</p>
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<div class="timestamp" onclick="openVideoSide('46:21', 'Boundary Term Analysis', 'proof', this)">
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<h3>Boundary Term Analysis</h3>
<p>For the boundary terms to vanish (making $L$ self-adjoint):</p>
<ul>
<li><strong>At singular endpoints:</strong> Since $p(c) = 0$, the boundary term automatically vanishes.</li>
<li><strong>At non-singular endpoints:</strong> Since both $y_1$ and $y_2$ satisfy the same real homogeneous boundary condition $\alpha_c y(c) + \beta_c y'(c) = 0$, and $\alpha_c, \beta_c$ are real, we can show that the Wronskian $W(y_1, y_2)$ evaluated at $c$ is zero.</li>
</ul>
<p>This is because the boundary conditions form a linear system with a non-zero determinant, forcing the Wronskian to be zero.</p>
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<div class="timestamp" onclick="openVideoSide('51:43', 'Historical Context', 'default', this)">
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<h3>Historical Context</h3>
<p>The history of Sturm-Liouville operators is fascinating - this is where analysis really started to get going. Before Cauchy and Weierstrass, it was "a jungle out there" with no rigorous foundations. The development of this theory shows how functional analysis was effectively born from these concrete problems.</p>
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