# Tagged Questions

For questions on eigenfunctions.

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### Solving non-homogenous PDE with forcing function (which diappears!) dependent only on time

Applying the method of eigenfunction expansion to the PDE $$u_t -c^2u_{xx}=F(t)$$ $$0<x<L, t>0$$ $$u(x,0)=f(x)$$ $$u_x(0,t)u_x(L,t)=0$$ for the homogenous part of this equation ($L[v(x,t)]=0$...
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### Resonance in wave equation

I have solved the non-homogenous equation by the method of eigenfunction expansion $$u_{tt} - c^2 u_{xx}=F(x)\sin(\omega t)$$ $$0<x<L, t>0$$ $$u(x,0)=u_t(x,0)=0$$ $$u(0,t)=u(L,t)=0$$ and got ...
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### Can “eigenvalues” in eigenfunction expansion be non-scalar?

This question is a bit nebulous but I don't have a particular example in mind... In general under certain assumptions, one can use eigenfunction expansion to represent an operator. For instance, for ...
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### Solving $\sin(\sqrt{\lambda}L) + \beta \cos(\sqrt{\lambda}L)\sqrt{\lambda} = 0$

I'm working with the ODE $$-\frac{d^2u}{dx^2}=\lambda u$$ and trying to find eigenvalues and eigenfunctions corresponding the boundary conditions $$u(0)=0, u(L)+\beta \frac{du}{dx}(L)=0$$ Assuming ...
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### variational principle for the principal eigenvalue

I am reading the proof of theorem 2 in chapter 6 Evan PDE. I have difficulty verifying the following part of the proof, i.e. 3 questions here. 1) The assumptions $u\in H_0^1(U)$ and $u\in L^2(U)=1$ ...
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### If we know Spec($M_1$) and Spec($M_2$), what could we say about Spec($M_1 \cup M_2$)?

Let two domains $M_1$ and $M_2$ (Dirichlet conditions). If we know the spectrum of the Laplacian on $M_1$ and $M_2$, what could we say about Spect($M_1 \cup M_2$)? Is there a theorem that might give ...
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### The bigger the domain, the smaller the first eigenvalue - $\lambda(M_2) \leq \lambda(M_1)$ on the Laplacian

I know it is probably a silly question, but is there anyone could help me to complete of the corollary $3.1$ of that document? I pass a lot of time to try understanding the problem, but I can't ...
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### Nullspace of strange operator

I have the following equation: $0=\frac{\partial}{\partial y}(e^{-\beta U(x,y)}\frac{\partial}{\partial y}(P(x,y,t)e^{\beta U(x,y)}))$ and would like to study its solvability (Fredholm) conditions (...
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### Weyl's asymptotic law for eigenvalue on the rectangle $D = \{0 < x < a, 0 < y < b \}$ - $N(\lambda) \geq \frac{\lambda ab}{4 \pi} - C \sqrt{\lambda}$

I have a few difficulties understanding the example on the rectangle in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $326$. I've managed to ...
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### Eigenvalues Eigenvectors and bases of eigenspace

I was given the matrix \begin{pmatrix} 3 & -5 & 4 \\ 2 & 0 & -3 \\ -1 & 2& -1 \end{pmatrix} I'm not sure what to do after I get $-x^3 + 2x^2 - 17x + 9=0$.
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### Eigenvalue and eigenvector problem

I am looking to get to the answer of Question 3 e). I successfully expand the determinant to -λ^3 + 6λ^2 + 3λ - 13 using the characteristic equation and this is where I get stuck. I tried rational ...
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### Generalized Hermite Function as eigenfunction of a differential operator

I'm going through this paper. The article defines function function $\phi_n^\mu(x)$ that is orthonormal on $L^2$ with measure $dm = dx$: \phi^\mu_n =\left(\frac{\gamma_\mu(n)}{\...
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### Minimal set of functions spanning the same space as a larger set

Apologies if this has been asked already; I'm not sure how to phrase it in a search. I have a set of functions $f_k$ with an inner product $\langle f_i,f_j\rangle$ (which I compute using Monte Carlo)....
I have to solve the following eigenvalue problem, i.e. find eigenvalues and eigenfunctions (some of you will notice that this is the Schrödinger equation): $$-\frac{\hbar^2}{2m}\left( \frac{\partial^... 1answer 11 views ### Triangle with the lowest laplacian eigenvalue under the Dirichlet boundary condition Let us fix the area of the triangle. Which triangle has the lowest Laplacian eigenvalue? The equilateral one? 0answers 15 views ### eigenfunction derivation for 2nd order ODE Any one help in the attached derivation. I am lost from eqn 4.13 till basically 4.24. Regards PDF file showing the proof steps 1answer 39 views ### How do solve this pde problem? EDIT: I know somehow, we end up with an equation relating the derivative of some coefficients to the rest of the stuff. I'm not sure where this equation, or even the constant that we use to get it, ... 1answer 40 views ### eigenvaue of Sturm Liouville problem Let the limit probem$$ \begin{cases} (P(x) y')' + q(x) y' + \lambda r(x) y=0\\ \alpha_0 y(0)+ \alpha_1 y'(0)\\ \beta_0 y(l) + \beta_1 y'(l) \end{cases}  with $\alpha_0^2 + \alpha_1^2 >0$ and $\... 0answers 44 views ### Eigenvalue of Integral Operator and Gamma Function$''$Prove that the following integral operator$ Ku(x) = \int_{0}^{ \infty } \ e ^{-xy} u(y) dy $has as eigenfunction the$ φ_α(x) = \sqrt {Γ(α)} x^{-α} + \sqrt {Γ(1-α)} x^{α-1} $for$ ...
Consider a densely defined unbounded operator $A_0:D(A_0)(\subset{H})\to H$ which has the following properties: 1- Symmetric, $\langle A_0x,y\rangle=\langle x,A_0y \rangle$ 2- Positive, \$\langle ...