1
vote
0answers
26 views

How to calculate Hill's discriminant?

I am currently reading this paper on Schrödinger operators see here. On page 6 and 7 they talk about Hill's discriminant and how this is connected with the spectral properties. They also show some ...
4
votes
0answers
247 views

Explain this step in lecture notes

The bounty offered is for the person that explains me how the author gets from equation 3.19 to equation 3.20 in these lecture see here. Normally I would agree that copying the relevant equation would ...
1
vote
1answer
67 views

how to solve the system of differential equations for this particle?

I'm trying to solve this problem A particle of mass m moves under the action of gravity on the inner surface of a paraboloid of revolution $x^2+y^2=az$ which assumed frictionless. Obtain the ...
2
votes
1answer
51 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
0
votes
1answer
42 views

Orthogonality of associated Legendre polynomials

Let $P_n(x)$ be the $n$-th degree Legendre polynomial. Let $k$ be a nonnegative integer less than or equal to both $n,m$. How to prove that $$ \int_{-1}^1 (1-x^2)^k D^kP_n(x) D^kP_m(x)\,dx = ...
1
vote
0answers
39 views

Meaning of this differentiation operators

I have been just reading this paper here: paper and was wondering how they carry out the differentiation in (4.9). In principle, this should be just the differentiation of 4.8 with the help of 4.7a. ...
2
votes
1answer
81 views

Bound on specific L2 function

Its stated in a paper by E. Lieb that this function is "clearly" in L2 $w(x)=|x|^{-1} - (g^2*|x|^{-1}*g^2)(x)$ with $g(x)=\xi^{3/4} \exp(-\pi \xi x^2/2)$ and therefore $||g||_2 =1$. It's ...
0
votes
0answers
32 views

Heisenberg uncertainty principle in D-dimensional

For Heisenberg uncertainty principle in D-dimensional there is $d^2$ in the formula.where does this additional term comes compared with the case of one dimensional?
2
votes
1answer
97 views

The Gaussian Integral

Hi I am trying to calculate the expected value of $$ \mathbb{E}\big[x_i x_j...x_N\big]=\int_{-\infty}^\infty x_ix_jx_k...x_N \exp\bigg({-\sum_{i,j=1}^N\frac{1}{2}x^\top_i A_{ij}x_j}-\sum_{i=1}^Nh_i ...
2
votes
1answer
65 views

Cauchy's problem. Equation of mathematical physics

$$U_{tt} = \Delta U + x^3 - 3xy^2$$ $$U|_{t=0} = e^x \cos y$$ $$U_t|_{t=0} = e^y \sin x$$ Help me, please, with solution of this equation. Can you prompt me algorithm to find the ...
2
votes
1answer
136 views

Relationship between adjoint operators, trace-class operators, compact operators and density operators in Quantum-Mechanics

I don't know much about Functional Analysis, but I was wondering about the following: In Banach spaces it is possible to define for every continuous opertor $T:X \rightarrow Y$ an adjoint Operator ...
2
votes
1answer
68 views

Closure of numerical range contains spectrum

Let $A: D(A) \subset \mathcal{H} \to \mathcal{H}$ be a densely defined operator on a Hilbert space $\mathcal{H}$ with adjoint operator $A^{*}$. Given that $D(A) = D(A^{*})$ I'm trying to show that the ...
7
votes
1answer
135 views

Positivity of the Coulomb energy in 2d

Let $$D(f,g):=\int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{1}{|x-y|}\overline{f(x)}g(y)~dxdy$$ with $f,g$ real valued and sufficiently integrable be the usual Coulomb energy. Under the assumption ...
1
vote
1answer
89 views

Bounded Self-adjoint Operator on Hilbert Space

I am trying to show that if $A$ is a bounded, self-adjoint and positive operator on a Hilbert space $H$, $0 \in \rho(A)$, the following inequality holds for all $x \in H$ with $\|x\| = 1$: ...
2
votes
3answers
206 views

What do physicists mean with this bra-ket notation?

In Quantum mechanics we said that $\langle x'|\psi \rangle = \psi(x)$, where $\langle \phi|\psi \rangle $ is the dot product in $L^2(\mathbb{C})$. I found out, that this is true, if you set x' to ...
4
votes
2answers
77 views

Solving $\int_{-\infty}^{\infty}\frac{x^2e^x}{(1+e^x)^2}dx$

I am attempting to use residues to solve $\int_{-\infty}^{\infty}\frac{x^2e^x}{(1+e^x)^2}dx$; the answer is $\frac{\pi^2}{3}$. I have tried to split $\frac{x^2e^x}{(1+e^x)^2}$ into two parts ...
0
votes
1answer
52 views

Suppose that $\int _0^1 f(x)v(x)=0$ for every $v \in C^{\infty}([0,1])$ for which $v'(0)=v(1/2)=0$. Show that $f(x)=0$ for all $x\in [0,1]$.

Suppose that $\int _0^1 f(x)v(x)=0$ for every $v \in C^{\infty}([0,1])$ for which $v'(0)=v(1/2)=0$. Show that $f(x)=0$ for all $x\in [0,1]$.Suggestion: take u to be the suitable cut off version of ...
2
votes
2answers
1k views

Commutator of $x$ and $p^2$

I have a question: If I have to find the commutator $[x, p^2]$ (with $p= {h\over i}{d \over dx} $) the right answer is: $[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$ But ...
1
vote
0answers
66 views

The equvalence of the virtual work and the Hamiltonian equations

I am reading Whittaker's Analytical Dynamics. This is chapter 10 *Hamiltonian Systems&. Paragraph 109 is Hamiltonian Systems & Their integral invariants. Whittaker starts with the Lagrangian ...
2
votes
0answers
73 views

Meaning of quasiperiodicity in classical KAM

I'm learning about the classical KAM theorem, and I can't quite infer precisely what the term "quasi-periodic solution" means in the theorem's statement. I'm reading the following introductory note: ...
1
vote
1answer
80 views

What are the structure constants for the algebra of quaternions? Show this algebra is associative.

What are the structure constants for the algebra of quaternions? Show this algebra is associative. How can I find the structure constants? I know that for an algebra $\mathscr{A}$ and basis ...
0
votes
1answer
31 views

surjectivity of a linear transformation and spanning

Okay, we have that $\{|a_i\rangle\}_{i=1}^n$ is a set of vectors spanning a vector space $V$. Also, $T\in L(V,W)$ is surjective, where $L(V,W)$ is the set of linear transformations (functionals) from ...
4
votes
0answers
70 views

Different functional brachystochrone

Until today I thought that $$ \int_0^b \sqrt{\frac{1+y'(x)^2}{2gy(x)}} dx$$ would be the only functional to derive the brachystochrone, but in the textbook Variational Methods in Mathematical Physics ...
1
vote
1answer
129 views

How to expand a fraction in powers of $z$ or $\dfrac{1}{z}$, and which to do, in determining Laurent series

I have a function $f(z)=\dfrac{12}{z(2-z)(1+z)}$, I'm trying to find the Laurent series for each of the three annuli. The singularities are at $z = 0$, $z = 2$, and $z = -1$, so I'm looking for three ...
0
votes
1answer
301 views

damped wave equation

For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation. My question is...why is the ...
5
votes
1answer
297 views

Expressing the wave equation solution by separation of variables as a superposition of forward and backward waves.

(From an exercise in Pinchover's Introduction to Partial Differential Equations). $$u(x,t)=\frac{A_0 + B_0 t}{2}+\sum_{n=1}^{\infty} \left(A_n\cos{\frac{c\pi nt}{L}}+ B_n\sin{\frac{c\pi ...