Questions tagged [sturm-liouville-theory]
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63
questions
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Chapter 2, Section 5 of Chavel's book “Eigenvalue In Riemann Geometry" is about the zero-point distribution of the derivatives of eigenfunctions
In Chapter 2, Section 5 of Chavel's book, regarding the Neumann eigenvalues of the Laplacian in space forms, how did Chavel determine that $T'_{l,j}$ has ($j-1$) zeros? I have consulted books on the ...
2
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0
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95
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On the decay of a supersolution of a Sturm Liouville eigenvalue problem
I am not really familiar with Sturm-Liouville theory, so possibly the answer to my question is rather trivial. Consider the SL problem
\begin{equation*}
L \varphi (x) : = \frac{d}{dx}\Big[ (x^2+1)^\...
1
vote
1
answer
72
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Spectrum below zero for $-\beta(x) \partial^2_x : L^2(\mathbb{R}) \to L^2(\mathbb{R})$
Let $\beta \in L^\infty(\mathbb{R} ; (0, \infty))$ be bounded from above and below by positive constants. Consider the self-adjoint operator $ -\beta^{-1} \partial^2_x : L^2(\mathbb{R}; \beta dx) \to ...
3
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1
answer
154
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Must solutions to the time-independent Schrodinger equation that have discrete or negative eigenvalues be square-integrable?
This is a very straightforward question (although the answer may not be!) about a ubiquitous textbook physics situation. I assume that the answer is probably well-known, but I asked it on both Math ...
0
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1
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69
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Orthogonality to a one parameter family of eigenfunctions
Let $\rho>0$ be a smooth realvalued function such that $\rho=1$ outside the unit interval $(-1,1)$. For each $t>0$, let us denote by $\{\lambda_n(t)\}_{n=1}^{\infty}$ and $\{\phi_n(t;x)\}_{n=1}^{...
1
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1
answer
151
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Existence for a singular Sturm-Liouville "eigenvalue" problem with non-homogeneous boundary condition
Consider the following singular Sturm-Liouville problem:
$$
-(r^{N - 1}h')' - r^{N - 1}c(r)h = \lambda r^{N - 3} h \text{ in } (0, 1), \qquad h(1) = \alpha
$$
where
$N \in \mathbb N$, $N \geq 3$;
$c(...
0
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0
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42
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Identity principle of solutions of SL-problems with matching values on open set
Situation (cut short): Corresponding solutions (by eigenvalue) of two given regular Sturm-Liouville problems with homogeneous Neumann BC, same spectrum but possibly distinct coefficient functions, &...
4
votes
1
answer
451
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Eigenvalues of Sturm–Liouville operator
Can we calculate the eigenvalues and eigenfunctions of the following operator in $W^{1,2}(\mathbb{R})$?
$$-\left(\frac{1}{\cosh^2x}\right)y''-\frac{2}{\cosh^4x}y=\lambda y.$$
1
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0
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71
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How does boundary perturbation affect the eigenvalues of differential equations?
There is a well-known procedure (at least to me) to compute how a small perturbation will affect the eigenvalues of a differential equation.
However, the method deals only with perturbing the ...
4
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165
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Techniques for showing non-degeneracy results (PDE)
Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
4
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4
answers
375
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Joint boundedness of solutions of a family of Sturm-Liouville ODE
Let us fix $0 \neq \lambda \in \mathbb{R}$. Let us consider the following ODE, on $[0,\infty)$: $$ y^{\prime \prime} (x) + \frac{r e^{-x}}{(1+e^{-x})^2} y(x) = -\lambda^2 y(x).$$ Here $r \ge 1$ is a ...
4
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0
answers
161
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Boundary conditions for singular Sturm-Liouville problem (boundary behavior of eigenfunctions)
I am not at all an expert in Sturm-Liouville theory, but I ended up on the following Singular Sturm Liouville problem:
\begin{equation}\label{1}
(1) \ \ \ \ \ \ \ \ \ \ \ y''(t)+\frac{\theta'(t)}{\...
0
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2
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694
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Orthogonality of Bessel function $\int_0^bxJ_a(\ell x)J_a(\ell' x)=0$ for $\ell\neq\ell'$
How do I show the above relation with Sturm-Liouville theory (assume the usual boundary conditions for the identity)? Here is what I have tried: if we start with
$$
\big(xJ_a'(\ell x)
\big)'+\left(\...
1
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1
answer
89
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Can we drop this annoying integral term to restore a Sturm-Liouville problem?
On $[0,1]$, let $f:[0,1]\to \mathbb{R}$ be positive and continuous, consider the equation:
$w''+w+\lambda f\cdot (w-\int_0^1 w)=0$($\lambda$ is an eigenvalue) subject to $w(0)=w(1)=0$. If the integral ...
3
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0
answers
158
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Negative eigenvalue for a periodic Sturm-Liouville problem
Let $f \in C^{\infty}([0, 2\pi])$ be a smooth function and consider the following periodic Sturm-Liouville problem:
$$\begin{cases} u''(x) + f(x)u(x) = - \lambda u(x) \\
u(0) = u(2\pi) \\
u'(0) = u'(2\...
2
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0
answers
41
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References on discrete Sturm-Liouville eigenvectors convergence
Let $ L : u_n \mapsto a_n u_{n + 1} + b_n u_n + a_{n - 1} u_{n -1} = \nabla ( a_n \Delta u_n ) + (b_n + a_n + a_{n - 1}) u_n $ be a discrete Sturm-Liouville operator, with $ \nabla u_n := u_{n + 1} - ...
1
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1
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105
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Sign of solution to (in)homogeneous linear ODE
Let $N \geq 3$ be a positive integer and $A >0, B \geq 0$ be two constants. Let $y: (0,\infty) \to \mathbf{R}$ be a solution to the following linear, inhomogeneous ODE: $y''(x) + \frac{N-1}{x} y'(x)...
1
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105
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Sturm-Liouville Problem: When does $w y^2$ vanish at a singular boundary point?
It is well known (e.g. Courant, Hilbert - Methods of Mathematical Physics) that solutions of the Sturm-Liouville problem on an interval $J=(a,b)$
\begin{equation} \tag{1}
\left(p y' \right)' - qy \; = ...
1
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0
answers
125
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Angular excitations and Schrodinger operators with radial potential in N-dimensions
Can someone please explain the following in mathematical language?
"First of all, angular excitations only push the energy up, never down, so it is enough to analyze spherically symmetric s-waves....
2
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208
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Fourier mode decomposition and eigenvalues of Schroedinger operators with radial potential in N-dimensions
In the study the stability of minimal hipersurfaces $\Sigma \subset \mathbb{R}^{N+1}$ one is lead to study the Morse index of a Schroedinger operator $J := - \Delta_g + |A|^2$ (usually called Jacobi ...
3
votes
0
answers
1k
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The Node Theorem - an argument from physics
The theorem on the number of zeros of a solution to a Sturm-Liouville equation is a well-know result in quantum mechanics. It doesn't seem to have a special name in the mathematics literature, but it ...
1
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1
answer
63
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Set of eigenvalues of the boundary problem
I'm looking for the results about the set of eigenvalues of boundary problem for differential equation
\begin{equation}
\bigl(p(x) u'(x; \lambda) \bigr)' + q(x) u(x; \lambda) = -\lambda w(x) u(x; \...
3
votes
0
answers
175
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Fourier transform of Green function and its derivative
Consider a real Sturm-Liouville operator $L$ on $[0,+\infty)$ and use the following notations : https://www.encyclopediaofmath.org/index.php/Titchmarsh-Weyl_m-function
Assume $a = 0$, $\alpha \in [0,\...
7
votes
1
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659
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Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials
There are some known criteria for the Sturm-Liouville Problem
\begin{equation} \tag{1}
\frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y
\...
2
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1
answer
131
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Common eigenvalues for two Sturm-Liouville problem
Does exist in literature any results concerning the common eigenvalues for the two eigenvalue problems of the form
$$y''(x)=\lambda^2 y(x)+\lambda a(x)y(x), \ x\in(0,1), $$$$z''(x)=\lambda^2 z(x)-\...
2
votes
2
answers
196
views
Eigenvalue problem of Schrodinger equation with polynomial growth potential on the real line
Consider the operator $L=-\frac{d^2}{dx^2}+q(x)$, where $q(x)$ is the potential with polynomial-type growth, say $|x|^s,s>1$. The eigenvalue problem
$$L\phi=\lambda\phi$$
where $\phi$ is in $L^2(\...
1
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0
answers
89
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Equivalence of solutions to Sturm-Liouville problem after translation of boundary conditions
My doubt is related to the equivalence between solutions of the following Sturm-Liouville problem:
\begin{equation}
r^{2}f''(r) + 2rf'(r) + \{\omega^{2}r^{2} - [j(j+1)-|q|^{2})]\}f(r)=0\,,\label{SL1}
\...
0
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2
answers
476
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Sturm Liouville differential equation and hypergeometric functions
I'm trying to understand how to solve this differential equation:
$ [z^2(1-z)\dfrac{d^2}{dz} - z^2 \dfrac{d}{dz} - \lambda] f(z) = 0 $
I know the solution is related to the hypergeometric function ...
1
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0
answers
48
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How to solve or analyse the smallest eigenvalue of 2 coupled 1st-order linear ODEs?
I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities
\begin{align}
-\mathrm{i} u'(x) +f^*(x) ...
1
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0
answers
58
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Sturm-Liouville-like Eigenproblem
Consider the piecewise-deterministic Markov process on $\mathbf{R}$ which
moves according to the vector field $\phi (x) = 1$,
experiences events at rate $\lambda(x) = 1$, and
at events, jumps ...
1
vote
0
answers
310
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Showing a differential operator is positive semidefinite
Let $R>\lambda>\chi$ be positive real constants and $\alpha$ be a real number. The following differential operator
\begin{multline}
\mathcal{L}g = -\frac{d}{d\xi}\left[(1-\xi^2)\frac{dg}{d\xi}\...
7
votes
1
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370
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Monotonicity of Schrödinger Eigenvalues
Let us consider the Schrödinger operator
$$
H_hf(x)=-\frac{d^2}{dx^2}f(x)+h(h\sin^2(x)-\cos(x))f(x)
$$
on $L^2[-\pi,\pi]$ with Neumann boundary conditions $f^\prime(\pm\pi)=0$. Here, $h\geq 0$ is a ...
2
votes
0
answers
97
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1D Schrödinger Equation with Measure-Valued Coefficients
I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following:
$$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}...
2
votes
1
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214
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Eigenvalues Sturm-Liouville Operator
Is the eigenvalue decomposition of the Sturm-Liouville operator
$$
Lf(x)=-f''(x)+h\sin(x)f'(x),\quad h>0,
$$
with Neumann boundary conditions $f'(-\pi)=f'(\pi)=0$ on the Hilbert space $L^2([-\pi,\...
4
votes
1
answer
144
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Monotonicity/Scaling of Sturm-Liouville Eigenvalues
Consider the regular Sturm-Liouville eigenvalue equation
$$
\frac{d}{dx}(p_t(x)f^\prime(x))=\lambda_t f(x)
$$
for $p_t\in\mathcal{C}^\infty([0,1])$ with Dirichlet boundary conditions on $[0,1]$. Here $...
2
votes
0
answers
149
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Limit circle/point of an ODE with finite eigenvalues
Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
4
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2
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570
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Orthogonal Polynomials and Sturm Liouville operators
Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define $L[u]=u''-xu'$, then the $n$-th order Hermite polynomial satisfies $...
1
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0
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133
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Sturm Liouville problem on entire line, substitution
Observe Sturm-Liouville problem on entire line
$$-(p(x)y'(x))' + l(x)y(x)= \lambda r(x)y(x), \hspace{3mm} -\infty<x<\infty
\tag{1} \label{1}$$
where $p(x)$ and $r(x)$ are positive on $\mathbb{R}$...
2
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0
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69
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One class Sturm-Liouville differential equation
Let $\phi_n(x), \psi_n(x)$ be solution Sturm-Liouville differential equation
$$p(x) y''(x) - 2n p'(x)y'(x)+2n(2n+1)y(x)=0$$
$$\phi_{n}(0)=0, \hspace{3mm} \phi'_{n}(0)=1;$$
$$\psi_{n}(0)=1, \hspace{...
1
vote
1
answer
426
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Completeness of the solutions to the Schrödinger Hydrogen Atom
I once did some work on using orthogonal function expansions for fitting 3D distribution functions. To ensure completeness over $L^2$ (which was considered sufficient even though technically a ...
1
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1
answer
119
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A property of a nonlinear ODE under periodic boundary conditions
Let $u_1,u_2 : (0,1)\to \mathbb{R}$, and given that $$u_1(0) = u_1(1),u_1'(0)= u_1'(1)$$ and $$u_2(0) = u_2(1),u_2'(0)=u_2'(1)$$ and also $$(|u_1'|u_1')' = \lambda_1|u_1|u_1$$ and $$(|u_2'|u_2')' = \...
2
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1
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300
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How to determine the spectrum from the diagonal Green's function
Let $L: L^2(\mathbb{R}) \supseteq Dom(L) \rightarrow L^2(\mathbb{R})$ be a densely defined closed operator. Assume that the resolvent admits an integral kernel (Greens function) $G$, i.e. for $z\in \...
3
votes
0
answers
105
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Convergence of Bessel (Sturm-Liouville) Expansions at the End Points
I have asked this question before on MSE but received no answer at all. So I assume that it is proper to ask it here. I am not a mathematician so my language may not be too precise, please correct me ...
3
votes
1
answer
158
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An indefinite integral containing functions that are solutions to a 2nd order linear ODE
I am trying to evaluate an indefinite integral of the form
$\int \frac{dz}{A u_1^2 + Bu_2^2 + Cu_1u_2}$
where $u_1$ and $u_2$ are two independent solutions to the ODE
$u'' + F(z)u = 0$
This ...
2
votes
1
answer
373
views
Sturm Liouville problems for non-classical orthogonal polynomials
It is known that for the classical orthogonal-polynomials there exist a set of Sturm Liouville problems. E.g. , the Hermite polynomial of order $n$ is a solution of $$y''(x) -xy'(x)+ny(x)=0 \, .$$
My ...
1
vote
0
answers
141
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References for the Sturm oscillation theorem
What is the most general form of the Sturm oscillation theorem?
So far I have only seen cases that treat either unbounded intervals or weighted $L^2$ spaces. I would be especially interested in ...
6
votes
3
answers
808
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Non-self adjoint Sturm-Liouville problem
I'm new to this site, but I felt the need to post when I recently came in to an ordinary differential equation/boundary value problem with this form:
$(1)- \frac{d^2 y}{d x^2} + \frac{m(m+1)} {x^2(...
2
votes
0
answers
70
views
Error bounds for eigenvalue expansion of the Mathieu equation
The Mathieu equation is an important eigenvalue problem in Mathematical Physics that is completely understood in its properties, although there is no "direct way" of expressing eigenvalues and ...
3
votes
2
answers
402
views
Schrödinger operators on a sphere
if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on ...
1
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1
answer
357
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Zeroes of Sturm-Liouville solutions as a function of the (complex) eigenvalue
Given the Sturm-Liouville type (time independent Schroedinger) equation
\begin{equation}
\frac{d^2 y}{d x^2} - \left(\mu + V(x)\right) y = \lambda \, y,\quad x \in \mathbb{R}
\end{equation}
where $V(...