User Disintegrating By Parts

16 votes

1 answer

3k views

How to use the Spectral Theorem to Derive $L^{2}(\mathbb{R})$ Fourier Transform Theory

asked Nov 5, 2014 at 7:07

16 votes

2 answers

800 views

Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

asked Feb 18, 2015 at 5:21

13 votes

1 answer

322 views

If $f\in C[0,1)$, $\int_{0}^{1}f^{2}(t)dt =\infty$, can one construct $g\in C[0,1)$ so $\int_{0}^{1}g^{2}dt < \infty$, $\int fgdt = \infty$?

asked Oct 15, 2015 at 5:26

11 votes

0 answers

256 views

What classical conditions give unique Laplace equation solutions on a half-plane?

asked Mar 15, 2015 at 17:38

7 votes

2 answers

5k views

How to show Plancherel's Theorem for Fourier Transform implies $L^2$ Transform Convergence.

asked May 3, 2016 at 20:05

6 votes

1 answer

2k views

Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$

asked Nov 6, 2014 at 19:00

6 votes

1 answer

135 views

Simple proof that $\|p(A)\|\le \sup_{|z|\le 1}|p(z)|$ for polynomials $p$ and $\|A\| \le 1$.

asked Oct 15, 2014 at 5:49

6 votes

1 answer

142 views

Show that $\|e^{tA}\| \le e^{t\|\Re (A)\|}$

asked Jun 15, 2014 at 21:33

5 votes

1 answer

146 views

How to show $e^{-x}$ is a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ in $L^{2}[0,\infty)$?

asked Oct 25, 2014 at 21:36

5 votes

1 answer

114 views

Is there a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ on $L^{2}[0,2\pi]$ with periodic conditions?

asked Oct 25, 2014 at 23:09

4 votes

1 answer

66 views

Is a function $f$ with $f(X)\perp (I-f)(X)$ necessarily linear?

asked Jun 15, 2014 at 20:26

4 votes

0 answers

116 views

Why do these Integration-by-Parts Evaluation Terms Vanish?

asked Nov 9, 2015 at 2:31

3 votes

1 answer

174 views

If $a_n \ge 0$ and $\sum_n a_n = \infty$, must $\sum_n \frac{a_n}{1+\sum_{k=1}^{n}a_k}$ also diverge?

asked Apr 25, 2016 at 20:23

3 votes

1 answer

217 views

How to see there exists const $C$ such that $\frac{d^{n-m}}{dx^{n-m}}(1-x^2)^n=C(1-x^2)^m\frac{d^{n+m}}{dx^{n+m}}(1-x^2)^n$

asked Dec 20, 2016 at 19:55

3 votes

1 answer

150 views

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

asked Jun 16, 2014 at 1:54

3 votes

1 answer

444 views

Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$

asked Aug 4, 2014 at 1:47

3 votes

2 answers

745 views

How to show Legendre Operator $L_{m}=-\frac{d}{dx}(1-x^{2})\frac{d}{dx}+\frac{m^{2}}{1-x^{2}}$ is Selfadjoint?

asked Mar 10, 2015 at 8:11

2 votes

0 answers

96 views

Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?

asked Jan 28, 2015 at 18:42

2 votes

1 answer

410 views

A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

asked Oct 18, 2014 at 21:48

2 votes

1 answer

168 views

How to show a Borel Operator Measure dilates to a Spectral Measure?

asked Oct 21, 2014 at 20:23

2 votes

1 answer

273 views

Finite-Dimensional Subspaces Invariant under Differentiation

asked Aug 8, 2014 at 13:33

2 votes

0 answers

48 views

How to show $\Im\{z\cot(z)\}$ is not $0$ in the first quadrant?

asked Jul 30, 2015 at 17:55

1 vote

0 answers

123 views

A Sturm-Liouville Problem with continuous spectrum and infinite point spectrum?

asked Oct 15, 2020 at 3:12

1 vote

2 answers

142 views

Completeness of Exponentials Using Complex Analysis

asked Oct 15, 2020 at 17:40

1 vote

0 answers

581 views

Derive Differentiation Identity for Gegenbauer Polynomial

asked Feb 23, 2014 at 20:23

0 votes

1 answer

142 views

Is this subspace dense in $L^{2}(\Omega,\mu)$

asked Jun 16, 2014 at 0:18

0 votes

0 answers

66 views

How Many Negative Eigenvalues of $-\frac{d^{2}}{dx^{2}}$ on $[0,L]$?

asked Sep 16, 2014 at 21:24

Stack Exchange Network

27 Questions

How to use the Spectral Theorem to Derive $L^{2}(\mathbb{R})$ Fourier Transform Theory

Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

If $f\in C[0,1)$, $\int_{0}^{1}f^{2}(t)dt =\infty$, can one construct $g\in C[0,1)$ so $\int_{0}^{1}g^{2}dt < \infty$, $\int fgdt = \infty$?

What classical conditions give unique Laplace equation solutions on a half-plane?

How to show Plancherel's Theorem for Fourier Transform implies $L^2$ Transform Convergence.

Proving Stone's Formula for Constructively obtaining the Spectral Measure for $A=A^\star$

Simple proof that $\|p(A)\|\le \sup_{|z|\le 1}|p(z)|$ for polynomials $p$ and $\|A\| \le 1$.

Show that $\|e^{tA}\| \le e^{t\|\Re (A)\|}$

How to show $e^{-x}$ is a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ in $L^{2}[0,\infty)$?

Is there a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ on $L^{2}[0,2\pi]$ with periodic conditions?

Is a function $f$ with $f(X)\perp (I-f)(X)$ necessarily linear?

Why do these Integration-by-Parts Evaluation Terms Vanish?

If $a_n \ge 0$ and $\sum_n a_n = \infty$, must $\sum_n \frac{a_n}{1+\sum_{k=1}^{n}a_k}$ also diverge?

How to see there exists const $C$ such that $\frac{d^{n-m}}{dx^{n-m}}(1-x^2)^n=C(1-x^2)^m\frac{d^{n+m}}{dx^{n+m}}(1-x^2)^n$

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

Selfadjoint Restrictions of Legendre Operator $-\frac{d}{dx}(1-x^{2})\frac{d}{dx}$

How to show Legendre Operator $L_{m}=-\frac{d}{dx}(1-x^{2})\frac{d}{dx}+\frac{m^{2}}{1-x^{2}}$ is Selfadjoint?

Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?

A Direct Proof of Representation Theorem for Positive Harmonic Functions in the Half Plane?

How to show a Borel Operator Measure dilates to a Spectral Measure?

Finite-Dimensional Subspaces Invariant under Differentiation

How to show $\Im\{z\cot(z)\}$ is not $0$ in the first quadrant?

A Sturm-Liouville Problem with continuous spectrum and infinite point spectrum?

Completeness of Exponentials Using Complex Analysis

Derive Differentiation Identity for Gegenbauer Polynomial

Is this subspace dense in $L^{2}(\Omega,\mu)$

How Many Negative Eigenvalues of $-\frac{d^{2}}{dx^{2}}$ on $[0,L]$?