All Questions
Tagged with sp.spectral-theory mp.mathematical-physics
58 questions
1
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0
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127
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Trace type convergence of the Laplacian on the box to the Laplacian on $\mathbb R^d$
Let $-\Delta \colon H^2(\mathbb R^d) \to \mathbb R^d$ be the (negative) Laplacian on the full space and $-\Delta_L$ the Laplacian acting on $L^2([-L,L]^d)$ with some boundary conditions making it self-...
- 61
6
votes
0
answers
113
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Schwartz kernel of spectral projection of Laplacian and integrated density of states
I'm reposting here a question I asked on MSE which did not receive an answer.
I am considering the Dirichlet Laplacian $\Delta$ on some smooth domain $U$. For now assume that $U$ is bounded, and later ...
- 251
12
votes
3
answers
2k
views
Why is resonance such a widespread phenomenon?
It is easy to mathematically describe the motion of a mass which is attached to a spring and also pushed around by a sinusoidal force. We get a differential equation of the form:
$$\frac{\mathrm{d}^2x}...
- 1,433
2
votes
1
answer
69
views
Spectral threshold effect: examples
I know that the effect of homogenization can be treated as a spectral threshold effect. I want to know more examples of spectral threshold effects in mathematical physics.
0
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0
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415
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Spectral theorem for commuting operators
Let $A_{1},...,A_{n}$ be densely defined self-adjoint operators on a separable Hilbert space $\mathscr{H}$. Suppose these have a common dense domain $D\subset \mathscr{H}$ and satisfy commutation ...
- 3,515
3
votes
0
answers
203
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On the spectrum of Fokker–Planck with linear drift
The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...
- 116
0
votes
0
answers
241
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About the proof of Lebesgue decomposition theorem for Hilbert spaces
Let $\mu$ be a Borel measure on $\mathbb{R}$. By the Lebesgue decomposition theorem, there exists measures $\mu_\text{pp}$, $\mu_\text{ac}$ and $\mu_\text{sing}$ such that $\mu = \mu_\text{pp}+\mu_\...
- 1,305
1
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0
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151
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Uniqueness of Borel functional calculus for unbounded self-adjoint operators
I was reading these short notes on the Borel functional calculus where the author discusses the uniqueness property of this calculus for both bounded and unbounded self-adjoint operators.
When it ...
- 1,305
2
votes
1
answer
456
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On a theorem of Carlson on the necessary and sufficient condition for a matrix to have $m$ real eigenvalues
Background: In the physics of open quantum systems the Lindbladian $\mathcal{L}$ governs the evolution of quantum states through the Lindblad master equation.
The Lindblad operator usually has ...
- 1,054
1
vote
1
answer
294
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Spectral perturbation theory of discrete spectra in presence of continuous spectrum
This is a 2 part question:
1). I am looking for a (hopefully accessible to beginning grad student who knows matrix perturbation theory) reference for doing concrete calculations of perturbed discrete ...
- 2,281
1
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0
answers
45
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Approximating spectra of (finite rank pertubations of) Laurent operators by spectra of (pertubations of) periodic finite operators
A tridiagonal matrix is a matrix which only has elements on three diagonals.
So for $\alpha, \beta, \gamma \in \mathbb{C}$ consider the bi-infinite tridiagonal Laurent operator $T$ with $\beta $ on ...
- 1,054
9
votes
1
answer
483
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Deriving Sommerfeld radiation condition from limiting absorption principle
For the Helmholtz equation
$$
-(\Delta + k ^2) u = f, \label{1}\tag{1}
$$
imposing the Sommerfeld radiation condition
$$
\lim_{r\to\infty} r ^{\frac{m-1}2} \left( u_r - i k u\right) = 0
$$
on $u$ ...
- 91
1
vote
0
answers
62
views
Class of spectral zeta functions whose analytic extension takes a particular form
In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...
- 105
2
votes
0
answers
69
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Mathematical reason for scatter states being special?
In infinite spectral theory, we have the discrete and continuous spectrum, which are called "bound" and "scatter states" in physics.
My understanding is, if $O \in B(H)$ is a self-...
- 569
0
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0
answers
210
views
Reed-Simon Vol. IV: Question regarding convergence of eigenvalues
I am reading through Chapter XIII.16 of Reed and Simon's Methods of Modern Mathematical Physics IV: Analysis of Operators about Schrödinger operators with periodic potentials. Since the topic is kind ...
- 91
4
votes
1
answer
155
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Resource on spectral theory for differential operators with symmetry groups
In Methods of Mathematical Physics IV by Reed and Simon, the authors cover Floquet theory in detail in Section XIII.16. On page 280, they note that
"A part of the analysis of [the periodic ...
- 854
2
votes
2
answers
389
views
Does this operator have a continuous, localized eigenfunction with negative eigenvalue?
I am looking at a class of operators
$$
L[f](x)=af_{xxxx}-bf_{xx}+\frac{d}{dx}(\delta(x)f_x)
$$ , a<0,b<0,
on the real line, where $\delta$ is Dirac-delta.
I am interested in ruling out the ...
- 116
2
votes
0
answers
145
views
Are Weyl sequences polynomially bounded?
Look at the Hilbert space $l^2( \mathbb{Z}) $ and let $A$ be a translation invariant band operator. I.e. if $\{ e_n \}_{n \in \mathbb Z} $ is the standard basis for $l^2( \mathbb{Z}) $ then it holds ...
- 1,054
3
votes
1
answer
151
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Commutation between integrating and taking the minimal eigenvalue
Let $S = (f_{ij})_{ij}$ be a $n \times n$ real symmetric matrix, with functions $f_{ij} \in L^1(\mathbb{R}^d,\mathbb{R})$ in it. We define $\left(\int u S \right)_{ij} = \int u S_{ij}$ as the ...
user140442
3
votes
0
answers
102
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Determining what happens to the spectrum of Schrödinger operator as boundary condition changes
I recently came across a problem in research, and I'm asking about it here after trying in math stack exchange with no luck.
Suppose I have a metric graph $G$ (or even a closed interval, to make ...
- 251
4
votes
0
answers
134
views
What is known about the density of states for the Anderson Model?
The Anderson Model is given by the random Hamiltonian (as an operator on $l^2(\mathbb{Z}^d)$)
$$
H_\omega = - \triangle + V(\omega)
$$
where $V(\omega) \mid x \rangle = \omega(x) \mid x \rangle$ ...
- 1,054
1
vote
0
answers
128
views
Angular excitations and Schrodinger operators with radial potential in N-dimensions
Can someone please explain the following in mathematical language?
"First of all, angular excitations only push the energy up, never down, so it is enough to analyze spherically symmetric s-waves....
- 141
4
votes
0
answers
155
views
Schrodinger operator with magnetic field: eigenvalues
Consider the self-adjoint operator on $L^{2}(\mathbb{R}^{N})$,
$$H=-\frac{1}{2}(\nabla-iA)^{2}+V,$$
where $A\in C^{\infty}(\mathbb{R}^{N}, \mathbb{R}^{N} )$, $V\in C^{\infty}(\mathbb{R}^{N})$, $V\...
- 41
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 ...
- 21.8k
2
votes
1
answer
165
views
Anderson localization for Bernoulli potentials on half-line
Anderson localisation for (discrete) Schrödinger operators with Bernoulli potentials on $l^2(\mathbb{Z})$ was proven in
https://link.springer.com/article/10.1007/BF01210702
I am wondering if there ...
- 223
6
votes
2
answers
539
views
Is there a reasonable notion of spectral theorem on a pre-Hilbert space?
I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...
- 269
2
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0
answers
59
views
Zero in the spectrum of an elliptic second order operator
This might be considered as a continuation of my previous question Spectrum of a linear elliptic operator
but is independent. I have another question on V. Gribov's paper "Quantization of non-Abelian ...
- 21.8k
2
votes
0
answers
162
views
Spectrum of a linear elliptic operator
In the paper in quantum fields theory by
Gribov,V.; (1978) "Quantization of non-Abelian gauge theories". Nuclear Physics B. 139: 1–19;
in Section 3 the author makes the following claim from PDE and ...
- 21.8k
4
votes
1
answer
221
views
Non-isolated ground state of a Schrödinger operator
Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\...
- 12.5k
1
vote
0
answers
61
views
Numerical computation of spectrum for operators on real line with "confining potential"
I am looking to understand the conditions under which one can expect "reasonably" accurate solution to leading eigenvalues/eigenvectors of a second order differential operator posed on the real line.
...
- 318
3
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1
answer
232
views
what is about the corresponding power series?
According to the papers The absolutely continuous spectrum of Jacobi matrices and these lecture notes:
periodicity ~ potential well or lattice (order)
lack of absolutely continued spectrum ~ Anderson ...
- 3,094
4
votes
1
answer
2k
views
Gutzwiller trace formula
I am reading A Proof of the Gutzwiller Semiclassical Trace Formula Using Coherent States Decomposition, Commmun. Math. Phys, 202, 463-480 (1990).
Gutzwiller trace formula says
where
and $g$ is a $C^...
- 587
1
vote
0
answers
61
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The ground state energy of an atom as a function of an external electric field
I think this question belongs to mathematical physics. The Hamiltonian of an N-electron atom in a homogeneous electric field is
$$ H =\left( \sum_{i=1}^N \frac{p_i^2}{2m } - \frac{Z e^2}{r_i} - E_z ...
- 193
6
votes
3
answers
916
views
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(...
- 71
5
votes
1
answer
2k
views
Eigenvalues of the D'Alembertian operator
My question about the spectral theory of the D'Alembertian operator on a Lorentzian manifolds (say the spacetime $M^{3+1}$) given by $$\square = -\partial_{t}^2 + \Delta$$ for the metric $g=(-+++)$. ...
- 67
2
votes
1
answer
136
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Proper domain for operators
in this paper on arxiv in equation 27, two operators
$$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$
and $$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + \frac{mx}{\sqrt{1-x^2}...
6
votes
1
answer
353
views
Domains of raising and lowering operators in QM
Let $H : \operatorname{dom}(H) \subset L^2(\Omega) \rightarrow L^2(\Omega)$, where $dom(H) \subset H^2(\Omega)$, $\Omega \subset \mathbb{R}$ should be a bounded open interval(so 1-d setting(!)) and $H$...
user54300
6
votes
3
answers
2k
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Some explanation about Dynin's formalism
I have seen this claim on the Wikipedia page for the Yang-Mills Millenium problem by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I ...
- 1,687
4
votes
1
answer
275
views
Asymptotic behavior of Schrödinger operators
I am currently dealing with $1$ or at most $2$-dimensional Schrödinger operators on compact domains. A classical result of spectral theory is the Weyl approximations for this operator
$H = -\Delta +V$....
user54300
2
votes
1
answer
279
views
Is the structure constant additive on connected components?
This is the reanimation of a question which already got an answer, that I did not fully understand. Coming back to it, after let it sit in a corner for some time, I keep not getting the point. I would ...
0
votes
1
answer
250
views
Spectrum of an angular-momentum related operator
Could someone please give me a reference for the eigenvalues and eigenstates of operators related to the angular momentum of a spinless, non-relativistic 2-D quantum particle?
In particular, I'm ...
- 321
2
votes
0
answers
103
views
What are the boundary asymptotics of harmonic symmetric transverse traceless rank-s tensors on $\mathbb{H}^n$ in the Poincare upper-half-space model? [closed]
This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf
In this paper some of its most important results about the asymptotics of symmetric traceless ...
- 1,893
6
votes
0
answers
200
views
Spectral theory for Dirac Laplacian on a funnel
I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...
- 313
5
votes
2
answers
757
views
Generalized basis
In quantum mechanics, people introduce the notion of "continuous basis" (I actually don't know the mathematical denomination of it). It is not a Schauder basis. I would like to know what could be a ...
- 189
15
votes
6
answers
3k
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Spectral theorem for self-adjoint differential operator on Hilbert space
I need a reference concerning a theorem that shows the following result, stated very roughly:
Given a self-adjoint differential operator densely defined on a Hilbert space, then the given Hilbert ...
5
votes
1
answer
254
views
Well defined Tensoring of spectral triples
Hi,
I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it.
Question: In connes standard model he takes ...
- 133
3
votes
1
answer
289
views
Avalanche Principle for higher dimensional unimodular matrices ?
Hello everyone,
I have a quick question for people working on quasi-periodic Schrodinger operators, Lyapunov exponents for Schrodinger cocycles or in other fields that might make them aware of this ...
- 33
2
votes
1
answer
301
views
Weyl quantization and convexity
Let $C$ be a convex subset of $\mathbb R^{2n}$ and $\mathbf 1_C$ be the characteristic function of $C$. Is it true that
$$\forall u\in\mathscr S(\mathbb R^n),\quad
\langle\mathbf 1_C^{Weyl}u,u\rangle\...
- 16.2k
8
votes
2
answers
583
views
Efficiently computing a few localized eigenvectors
Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$.
The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...
- 817
4
votes
3
answers
643
views
Traceless GUE : Four Centered Fermions
The proof of the Wigner Semicircle Law comes from studying the GUE Kernel
$$ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...
- 22.8k