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What's the appropriate notion of a Unitary representation of a Lie algebra?

Here Lie algebras/groups are real. The most straightforward definition might be: Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...
Alex Zorn's user avatar
  • 922
12 votes
0 answers
478 views

What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...
Dan Ramras's user avatar
  • 8,803
8 votes
0 answers
221 views

Density of odd and even eigenstates of an integral operator

Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function. Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...
Alex's user avatar
  • 81
7 votes
0 answers
269 views

Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$

I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by: $$ Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy $$ This operator ...
martin tassy's user avatar
7 votes
0 answers
501 views

intuitive connection between The KdV equations and the Virasoro bott group

I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group....
R Mary's user avatar
  • 979
6 votes
0 answers
159 views

Identification of Fock space and the $L^2$ space of tempered distributions

Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the ...
CBBAM's user avatar
  • 721
6 votes
0 answers
123 views

Can two eigenfunctions be almost linearly dependent in a region?

Consider the Schrödinger operator $H=-\Delta+|x|^a$ on $\mathbb{R}$, where $a>0$. Since the potential is growing at $\infty$, we have compact resolvent thus the eigenvalues are discrete and tend to ...
Tomas's user avatar
  • 879
6 votes
0 answers
290 views

Two questions about Fock spaces

Let $\mathscr{H}$ be a complex Hilbert space and denote $\mathscr{H}_{n}$ the tensor product $\overbrace{\mathscr{H}\otimes\cdots\otimes\mathscr{H}}^{\text{n}}$. Denote by $\Pi_{\pm}$ the projection ...
JustWannaKnow's user avatar
6 votes
0 answers
262 views

Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
Tom LaGatta's user avatar
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6 votes
0 answers
411 views

Birth-Death Process associated with Orthogonal Polynomials

I have read in various places the following objects are related: orthogonal polynomials birth-death processes Lattice paths continued fractions After a lot of searching online, I found sketches ...
john mangual's user avatar
  • 22.8k
5 votes
0 answers
110 views

Clifford algebras in the context of locally convex topological vector spaces

Suppose given a locally convex Hausdorff topological vector space $V$ over $\mathbb R$ and a continuous, symmetric, bilinear map $q:V\otimes V\to \mathbb R$, where the tensor product is the completed ...
Eugene Rabinovich's user avatar
5 votes
0 answers
105 views

Paving property

In their famed paper (https://arxiv.org/abs/math-ph/0011053), Bourgain and Goldstein conjecture what they call the paving property: Let $H_{jk}=\delta_{j,k+1}+\delta_{j,k-1}+v(\theta+j\omega)\delta_{...
Eduard Tetzlaff's user avatar
5 votes
0 answers
611 views

unitary equivalence

Let $U$ be the bilateral shift operator in $l^2(Z)$, and let $V$ stand for a rotation on an irrational angle $\alpha$ in $L^2(T, \mu)$, where $T$ is a circle with a rotation-invariant Lebesgue measure ...
Pavel Kalouguine's user avatar
5 votes
0 answers
616 views

Lebesgue measure on Frechet space?

It is well known that there are no Lebesgue measures on infinite-dimensional Banach spaces (see e.g. http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure). However, I couldn'...
Vanessa's user avatar
  • 1,368
4 votes
0 answers
141 views

Generalizing Kato-Seiler-Simon-type inequalities to diamagnetic operators

I recently learned about estimates one can perform with operators on $L^2(\mathbb{R}^n)$ given as $f(x)g(-i\nabla)$, see Chapter 4 in Trace Ideals and their Applications by Professor Barry Simon (the ...
garserdt216's user avatar
4 votes
0 answers
119 views

Relationship between canonical commutation relations and projective representations?

$\DeclareMathOperator\CCR{CCR}\DeclareMathOperator\Im{Im}\DeclareMathOperator\PU{PU}$Let $V$ be a real vector space equipped with an antisymmetric bilinear form $\omega$. Recall that there is a $C^\...
Tim Campion's user avatar
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4 votes
0 answers
93 views

Conditions on the Hamiltonian of a classical system that yeild essentially self-adjoint quantum Hamiltonian

What are the conditions on the Hamiltonian of a classical system that under these conditions the quantum Hamiltonian obtained via Weyl quantization will be essentially self-adjoint in $L_2(\mathbb{R}^...
Glinka's user avatar
  • 381
4 votes
0 answers
70 views

Estimate the composition of a bounded multiplier with a trace class operator

Let $T$ be a trace class operator on $\ell^2 (\mathbb{N})$. Let $A$ be a multiplier on $\ell^2 (\mathbb{N})$ defined by a sequence $a=(a_n)_{n\in\mathbb{N}}$ in $\ell^{\infty} (\mathbb{N})$. That is, ...
Chuwei Zhang's user avatar
4 votes
0 answers
297 views

Which orbits of a separable representation of the infinite unitary group are closed?

Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following: Is it true that all ...
Michał Oszmaniec's user avatar
3 votes
0 answers
67 views

Effective action of unbounded operators on subspaces outside their domains of definition

Consider a densely defined, self-adjoint operator $$ H: \mathcal{D} \rightarrow \mathscr{H}. $$ Assume for simplicity that $H$ is nonnegative. We want to effectively restrict this operator $H$ to a ...
Qualearn's user avatar
  • 133
3 votes
0 answers
219 views

Can any POVM be induced by a quantum instrument?

I suspect this is the obvious result of something in operator algebras, but that's far outside my field. Recall that a projection-valued measure is a map $E$ from a sigma-algebra $\mathcal{F}$ on ...
Yonah Borns-Weil's user avatar
3 votes
0 answers
463 views

Orthonormal basis of eigenvectors of Hamiltonian - Is there any theorem justifying the physicist approach?

In his book The Principles of Quantum Mechanics, Dirac states: "We call a real dynamical variable whose eigenstates form a complete set an observable." To Dirac, any observable has a ...
MathMath's user avatar
  • 1,305
3 votes
0 answers
115 views

Linearized NLS/GP around a soliton and the spectrum of the evolution operator

I apologize if this has been asked before but so far I haven't found it anywhere. Consider the Nonlinear Schrödinger equation with a potential (i.e. Gross- Pitaevskii) in $\mathbb{R}^{d}$ $$i\Psi_{t} =...
Taotology's user avatar
3 votes
0 answers
102 views

Determining what happens to the spectrum of Schrödinger operator as boundary condition changes

I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck. Suppose I have a metric graph $G$ (or even a closed interval, to make ...
GSofer's user avatar
  • 251
3 votes
0 answers
127 views

Rigorous stability analysis of infinite dimensional ODEs : How to bound the tails?

My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider, $\dot{x}=Ax$, where $x$ is the infinite dimensional ...
Piyush Grover's user avatar
3 votes
0 answers
235 views

Chern number of projection-Topological magic in physics

I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...
Ben Curnow's user avatar
3 votes
0 answers
210 views

Meromorphic continuation of resolvent of free Laplacian on homogeneous Sobolev space

Let $n \ge 2$. Set $\dot{H}^1(\mathbb{R}^n)$ to be the homogeneous Sobolev space, defined as the Hilbert completion of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm $\| \varphi \|^2_{\dot{H}^1} \...
JZS's user avatar
  • 481
3 votes
0 answers
57 views

Integration of Weyl operators multiplied by quasifree state over a symplectic space

I am reading the book "An invitation to the Algebra of Canonical Commutation Relations" by Denes Petz. It is freely available for download here. In Chapter 9, he defines the Lebesgue measure on a ...
Tiju Cherian John's user avatar
3 votes
0 answers
94 views

Multiplicativity of $\zeta$-function regularized determinant

Let $A$ be a selfadjoint elliptic differential operator on a compact manifold. In mathematical physics and differential topology one often defines its determinant using the $\zeta$-function ...
asv's user avatar
  • 21.8k
3 votes
0 answers
188 views

Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?

This question is related to the following question Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)? A couple of authors have observed that composing a ...
jzadeh's user avatar
  • 265
3 votes
1 answer
310 views

Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?

Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding ...
Michał Oszmaniec's user avatar
2 votes
0 answers
60 views

Basis vectors using anti-commuting operators?

Let $V$ be a finite-dimensional inner product space. Suppose $A_{1},...,A_{N}$ are anti-commuting operators, meaning that these are linear operators on $V$ that satisfy: $$A_{i}A_{j}+A_{j}A_{i} = A_{i}...
MathMath's user avatar
  • 1,305
2 votes
0 answers
172 views

AQFT from a Lagrangian

In physics, the fundamental description of physical theories frequently revolves around the concept of a Lagrangian. My expertise encompasses diverse algebraic formulations within the domain of ...
Gabriel Palau's user avatar
2 votes
0 answers
463 views

Segal's axioms for CFT

In Segal's papers about Conformal Field theory, https://www2.math.upenn.edu/~blockj/scfts/segal.pdf, in section $1$, he describes the evolution of a system (a string moving about in a manifold $M$) by ...
Guillermo García Sáez's user avatar
2 votes
0 answers
92 views

Linearization stability condition

The following is a theorem from Fischer and Marsden's 1975's paper: Linearization stability of nonlinear PDEs. Theorem. Let $X, Y$ be Banach manifolds and $\Phi: X \rightarrow Y$ be $C^1$. Let $x_0 \...
Gordhob Brain's user avatar
2 votes
0 answers
172 views

Fourier transform harmonic oscillator eigenstates

The normalized eigenfunctions of the quantum harmonic oscillator are $$\psi_{n}(x)= \frac{1}{\sqrt{2^n n!}} e^{-x^2/2}H_n(x),$$ where $n \in \mathbb N_0$ and $H_n$ is the $n$-th Hermite polynomial, ...
Pritam Bemis's user avatar
2 votes
0 answers
85 views

Pullbacks of LCS-valued distributions

Suppose $X$ is a locally convex space. Since the distributions $\mathcal{D}'\!(M)$ ($M$ a manifold) are a nuclear space, there is a canonical meaning to the topological tensor product $X\,\widehat{\...
J_P's user avatar
  • 439
2 votes
0 answers
173 views

Product of Heavisides: calculus vs Fourier transform vs wavefront set

I decided to ask this question here, since I did not get any answer from MSE and perhaps this topic is somewhat far from MSE's topics. I am following the paper here. I am trying to understand how to ...
Evangeline A. K. McDowell's user avatar
2 votes
0 answers
118 views

What is the justification for using Wiener integrals to integrate over a space of differentiable functions?

In the literature on stiff/semiflexible polymer chains modelled as continuous chains rather than as discrete links, the partition function (among other things) is taken to be an integral over the ...
Harmenszoon's user avatar
2 votes
0 answers
195 views

Trouble understanding Lax method for KDV equation for inverse scattering method

I am trying to learn the Lax pair condition on my own so that I can eventually learn the inverse scattering method. I am following a paper by Tuncay Aktosun ("Inverse scattering transform and the ...
Will_Phys4's user avatar
2 votes
0 answers
45 views

Additivity of squared Schatten $p$-norm with respect to spatial partition

Consider a Hilbert-Schmidt operator $A$ on $L^2(\mathbb R^d)$ with integral kernel $A(x,y)$. Let $\Omega\subset \mathbb R^d$ and $1_{\Omega}(x)$ denote its characteristic function as well as the ...
user271621's user avatar
2 votes
0 answers
67 views

Reflection positivity on weighted $L^2$-spaces

Denote by $(t, x_{1}, \ldots, x_{d-1})$ the coordinates of $x \in \mathbb{R}^{d}$ and set $$\mathbb{R}^{d}_{+}=\left\{t, x_{1}, \ldots, x_{d-1} \in \mathbb{R}^{d}|t > 0\right\}. $$ Write $\theta$ ...
S.Z.'s user avatar
  • 505
2 votes
0 answers
78 views

How to compute the functional derivative of the following functional encountered in electrical impedance tomography?

Note: I have raised this question in Mathematical stack exchange but it received no attention. That is why I proceed to here to ask this question. Please tell me if this is not appropriate, thank you....
Ken Hung's user avatar
  • 161
2 votes
0 answers
145 views

Are Weyl sequences polynomially bounded?

Look at the Hilbert space $l^2( \mathbb{Z}) $ and let $A$ be a translation invariant band operator. I.e. if $\{ e_n \}_{n \in \mathbb Z} $ is the standard basis for $l^2( \mathbb{Z}) $ then it holds ...
Frederik Ravn Klausen's user avatar
2 votes
0 answers
61 views

Uniqueness of solution to Cauchy problem with quadratic nonlinearity

Consider the non-linear differential operator $$\mathfrak{L}: \ C^2((0,T)\times\mathbb{R}^2)\ni\varphi\equiv\varphi(t,x,y) \, \mapsto \, \partial_x^2\varphi + (\partial_x\varphi)^2.$$ For $U\subset\...
cts12's user avatar
  • 51
2 votes
0 answers
158 views

Lippmann-Schwinger equation for the Coulomb potential

Let $H=H_0+V$ be a Hamiltonian on $\mathbb{R}^3$ where $H_0=-\frac{\Delta}{2m}$ is the free Hamiltonian and $V$ is a potential. Let us assume first that $V$ decays sufficiently fast at infinity and ...
asv's user avatar
  • 21.8k
2 votes
0 answers
122 views

Reference on iterated integrals against projection valued measures

I know (to some extent) how integration over $\mathbb{R}$ of a Borel-measurable function against a projection-valued measure works. Recently while reading a paper I came across calculations in which ...
Cabbage's user avatar
  • 183
2 votes
0 answers
51 views

What restrictions on the form of an integral equation have a unique solution f=0?

We're stuck on the following question in a problem relevant to a physics paper on AdS/CFT that we are working on. Given a Fredholm equation of second kind with the form $f$+$\int_D K f\,dx = 0$, where ...
Ning Bao's user avatar
2 votes
0 answers
218 views

Existence of solutions to time-dependent Schrödinger equations

I would like to know what is known about evolution equations of the form $$iy'(t)=H_0y(x,t)+u(t)V(x)y(x,t)$$ and $y(0)=y_0 \in D(H_0)$ where $V$ is not a bounded operator, but an unbounded one, $u \...
Landauer's user avatar
  • 173
2 votes
0 answers
110 views

Boltzmann equation and the meaning of the marginals

I have a question related to the boltzmann equation and the meaning of the marginals. Let me first introdiuce the model and notation : (see for example https://arxiv.org/abs/1208.5753) We study ...
RaphaelB4's user avatar
  • 4,361