Active questions tagged sp.spectral-theory - MathOverflowmost recent 30 from www.4124039.com2019-05-23T16:01:51Zhttp://www.4124039.com/feeds/tag?tagnames=sp.spectral-theory&sort=newesthttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://www.4124039.com/q/3322700Show convergence of a sequence of resolvent operators0xbadf00dhttp://www.4124039.com/users/918902019-05-23T06:15:12Z2019-05-23T06:15:12Z
<p>Let</p>
<ul>
<li><span class="math-container">$E$</span> be a locally compact separable metric space</li>
<li><span class="math-container">$(\mathcal D(A),A)$</span> be the generator of a strongly continuous contraction semigroup on <span class="math-container">$C_0(E)$</span></li>
<li><span class="math-container">$E_n$</span> be a metric space for <span class="math-container">$n\in\mathbb N$</span></li>
<li><span class="math-container">$(\mathcal D(A_n),A_n)$</span> be the generator of a strongly continuous contraction semigroup on<span class="math-container">$^1$</span> <span class="math-container">$B(E_n)$</span></li>
<li><span class="math-container">$\pi_n:E_n\to E$</span> be continuous and <span class="math-container">$$\iota_nf:=f\circ\pi_n\;\;\;\text{for }f\in C_0(E)$$</span> for <span class="math-container">$n\in\mathbb N$</span></li>
</ul>
<blockquote>
<p>Let <span class="math-container">$\lambda>0$</span> and <span class="math-container">$f\in C_0(E)$</span>. Assume<span class="math-container">$^2$</span> <span class="math-container">$$\left|\left(R_\lambda(A_n)\iota_nf\right)(x_n)-\left(R_\lambda(A)f\right)(x)\right|\xrightarrow{n\to\infty}0\tag1$$</span> for all <span class="math-container">$x_n\in E_n$</span>, <span class="math-container">$n\in\mathbb N$</span>, and <span class="math-container">$x\in E$</span> with <span class="math-container">$\pi_n(x_n)\xrightarrow{n\to\infty}x$</span>. Are we able to conclude <span class="math-container">$$\left\|R_\lambda(A_n)\iota_nf-\iota_nR_\lambda(A)f\right\|_\infty\xrightarrow{n\to\infty}0;\tag2$$</span> at least under suitable further assumptions (e.g. compactness of <span class="math-container">$E$</span>)?</p>
</blockquote>
<p>Note that the result holds if <span class="math-container">$E_n=E$</span> for all <span class="math-container">$n\in\mathbb N$</span>, <span class="math-container">$E$</span> is compact and <span class="math-container">$\iota_n$</span> is the identity for all <span class="math-container">$n\in\mathbb N$</span>: <a href="https://math.stackexchange.com/q/3139957/47771">https://math.stackexchange.com/q/3139957/47771</a>.</p>
<p>We may note that by contractivity, <span class="math-container">$(0,\infty)$</span> is contained in the resolvent sets of <span class="math-container">$(\mathcal D(A_n),A_n)$</span>, <span class="math-container">$n\in\mathbb N$</span>, and <span class="math-container">$(\mathcal D(A),A)$</span>. Moreover, <span class="math-container">$$\left\|R_\lambda(A_n)\right\|,\left\|R_\lambda(A)\right\|\le\frac1\lambda\;\;\;\text{for all }n\in\mathbb N.\tag3$$</span> This might be crucial.</p>
<hr>
<p><span class="math-container">$^1$</span> If <span class="math-container">$S$</span> is a set, let <span class="math-container">$B(S)$</span> denote the space of bounded functions from <span class="math-container">$S$</span> to <span class="math-container">$\mathbb R$</span> equipped with the supremum norm.</p>
<p><span class="math-container">$^2$</span> If <span class="math-container">$(\mathcal D(B),B)$</span> is a bounded linear operator on a Banach space and <span class="math-container">$\lambda$</span> is a regular value of <span class="math-container">$(\mathcal D(B),B)$</span>, let <span class="math-container">$R_\lambda(B)$</span> denote the <a href="https://en.wikipedia.org/wiki/Resolvent_set" rel="nofollow noreferrer">resolvent operator</a> of <span class="math-container">$(\mathcal D(B),B)$</span>.</p>
http://www.4124039.com/q/3319961Strong Differentiability of Spectral ProjectionsLR235http://www.4124039.com/users/1408602019-05-20T09:48:41Z2019-05-20T09:48:41Z
<p>Let <span class="math-container">$H$</span> be a Hilbert space and <span class="math-container">$W$</span> be a dense subspace, equipped with a different norm that turns it into a Hilbert space. Let <span class="math-container">$(A(t))_{t\in[0,T]}$</span> be a family of Operators in <span class="math-container">$B(W,H)$</span> (bounded operators from <span class="math-container">$W$</span> to <span class="math-container">$H$</span>) that are self-adjoint with discrete spectrum as unbounded operators in <span class="math-container">$H$</span> with domain <span class="math-container">$W$</span>. Assume that <span class="math-container">$A(t)$</span> is differentiable as a function of <span class="math-container">$t$</span> in the strong topology on <span class="math-container">$B(W,H)$</span> and that 0 is not in the spectrum of <span class="math-container">$A(t)$</span> for any <span class="math-container">$t$</span>. Does this imply that the positive spectral projection of <span class="math-container">$A(T)$</span>, i.e. <span class="math-container">$\chi_{[0,\infty)}(A(t))$</span>, is differentiable in <span class="math-container">$t$</span> with respect to the strong topology on <span class="math-container">$B(H,H)$</span>? Does someone know a reference where a statement like this might be found?</p>
http://www.4124039.com/q/33187010Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifoldMax Schattmanhttp://www.4124039.com/users/1257902019-05-18T11:37:28Z2019-05-19T02:02:24Z
<p>Let <span class="math-container">$(M,g)$</span> be a compact Riemannian manifold, and let <span class="math-container">$\Delta_g$</span> be its Laplace-Beltrami operator. A "well-known fact" is that the eigenvalues of <span class="math-container">$\Delta_g$</span> have finite multiplicity and tend to infinity. What is the easiest/simplest way to estblish this fact? By simplest, I mean an abstract, formal, approach to the problem, not too heavily based on technical calculation. In the same line of thought, is there a philosophical/intuitive reason to see why the spectrum should behave like this? </p>
http://www.4124039.com/q/3315302How does $E$ closed follow from the upper semicontinuity of the spectrum?DJShttp://www.4124039.com/users/609132019-05-14T20:10:37Z2019-05-15T20:28:37Z
<p>Let <span class="math-container">$f$</span> be an analytic function for a domain <span class="math-container">$D$</span> of <span class="math-container">$\mathbb{C}$</span> into a Banach algebra <span class="math-container">$A$</span>. Suppose that, for all <span class="math-container">$\lambda \in D$</span>, <span class="math-container">$\text{Sp}f(\lambda)$</span> is finite or a sequence converging to <span class="math-container">$0$</span>.
Suppose that <span class="math-container">$\mu \neq 0$</span> and <span class="math-container">$\mu \in \text{Sp}f(\lambda_0)$</span> for some <span class="math-container">$\lambda_0 \in D$</span>.</p>
<p>Consider the set <span class="math-container">$E = \{ \lambda \in D: \mu \in \text{Sp}f(\lambda) \}$</span>.</p>
<p>B. Aupetit mentions in a proof he writes for Theorem 3.4.26 in his book <em>A Primer on Spectral Theory</em>, that this set <span class="math-container">$E$</span> is closed by the upper semicontinuity of the spectrum. I am struggling to see why this is true.</p>
<p>Can anyone please point me in the right direction as to how I can show that <span class="math-container">$E$</span> is closed by the upper semicontinuity of the spectrum?</p>
http://www.4124039.com/q/3148523Asymptotic behaviour of principal eigenfunctions and large deviationsleo monsaingeonhttp://www.4124039.com/users/337412018-11-08T16:00:25Z2019-05-14T06:18:39Z
<p>Dear Math Overflowers,</p>
<p>I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more details if needed.</p>
<p>I'm working in the 1D domain <span class="math-container">$\Omega=(0,1)$</span>, I have a fixed weight <span class="math-container">$\Theta(x)$</span> which is positive in the interior and vanishes linearly at the boundaries, typically <span class="math-container">$\Theta(x)=x(1-x)$</span>.
Let <span class="math-container">$\kappa>0$</span> be a small viscosity parameter, and finally consider a fixed potential <span class="math-container">$\Phi(x)$</span> smooth up to the boundary.</p>
<p>It is not too difficult to check that the principal eigenvalue problem
<span class="math-container">$$
\left\{
\begin{array}{ll}
-\kappa\partial_{xx}^2 (\Theta u) -\partial_x(\Theta u \partial_x\Phi) =\lambda u & \mbox{in }\Omega \\
\Theta u=0 & \mbox{on }\partial\Omega
\end{array}
\right.
$$</span>
and its adjoint
<span class="math-container">$$
\left\{
\begin{array}{ll}
-\kappa\Theta\partial_{xx}^2 v +\Theta \partial_x\Phi \partial_x v =\lambda v & \mbox{in }\Omega \\
v=0 & \mbox{on }\partial\Omega
\end{array}
\right.
$$</span>
are well-posed (note however that <span class="math-container">$\Theta$</span> vanishes at the endpoints, so this is perhaps not completely trivial -- but true nonetheless).
Moreover, I choose to normalize my eigenfunctions <span class="math-container">$u,v>0$</span> in such a way that
<span class="math-container">$$
\int_0^1 u=\int_0^1 uv =1.
$$</span>
Emphasizing now the dependence on <span class="math-container">$\kappa$</span>, let me define the probability measure
<span class="math-container">$$
\pi_\kappa:=u_\kappa(x)v_\kappa(x) dx.
$$</span></p>
<blockquote>
<p><strong>Question:</strong> is there any standard way to prove that, in the vanishing viscosity limit <span class="math-container">$\kappa\to 0$</span>, the sequence <span class="math-container">$(\pi_\kappa)_{\kappa>0}$</span> satisfies a <a href="https://en.wikipedia.org/wiki/Large_deviations_theory#Formal_definition" rel="nofollow noreferrer">Large Deviation Principle</a> with speed <span class="math-container">$\frac 1\kappa$</span> and rate function <span class="math-container">$\Phi(x)$</span> precisely given by my prescribed potential?</p>
</blockquote>
<p>I suspect that the <a href="https://en.wikipedia.org/wiki/Freidlin%E2%80%93Wentzell_theorem" rel="nofollow noreferrer">Freidlin-Wentzell theory</a> should help, but I am not as acquainted with probability theory as I would like to be...
Also, before trying brute force and spending some time trying to get fine information on each eigenfunction <span class="math-container">$u_\kappa$</span> and <span class="math-container">$v_\kappa$</span> separately (in order to take the product in the end), I am wondering if there might be a lazy way around and only talk of the product itself? At the PDE level both the non self-adjointness and the degeneracy of the diffusion (<span class="math-container">$\Theta$</span> vanishes) make the problem tricky. I have no clue about what's going on at the probabilistic level, but this looks like some product of "forward <span class="math-container">$\times$</span> backward" Kolmogorov operators/eigenfunctions, so perhaps there is some literature out there?</p>
http://www.4124039.com/q/3202504Lower estimate of the minimal eigenvalue of a Hamiltonianorbitshttp://www.4124039.com/users/161832019-01-06T23:41:51Z2019-05-07T19:03:50Z
<p>Consider a linear operator <span class="math-container">$H\colon L^2(\mathbb{R}^3)\to L^2(\mathbb{R}^3)$</span> given by
<span class="math-container">$$H(\psi)(x):=-\Delta\psi(x)+V(x)\cdot \psi(x),$$</span>
where <span class="math-container">$V\colon \mathbb{R}^3\to \mathbb{R}$</span> is a continuous (or smooth) non-negative function. If necessary, one may assume <span class="math-container">$\lim_{|x|\to \infty}V(x)=+\infty$</span>.</p>
<blockquote>
<p>Are there known estimates from below of the minimal eigenvalue of <span class="math-container">$H$</span> in terms of <span class="math-container">$V$</span>?</p>
</blockquote>
<p>(Of course, the trivial estimate is by 0.)</p>
<p>I am not an expert and may not aware even of the most standard results.</p>
http://www.4124039.com/q/3307701Bounds on spectral radius using chromatic numberLearnmorehttp://www.4124039.com/users/1402632019-05-05T09:04:21Z2019-05-05T09:04:21Z
<p>I am struggling with this question:</p>
<blockquote>
<p>If I have a connected graph <span class="math-container">$G$</span> on <span class="math-container">$n$</span> vertices and <span class="math-container">$m$</span> edges with chromatic number <span class="math-container">$d$</span> then how can I give a bound(lower and upper) on its spectral radius in terms of its chromatic number.</p>
</blockquote>
<p>There are some properties on the largest eigen value of adjacency matrix <span class="math-container">$A$</span> of a graph <span class="math-container">$G$</span>.</p>
<p>If a Graph <span class="math-container">$G$</span> is connected then the largest eigen value of <span class="math-container">$A$</span> say <span class="math-container">$\rho(A)$</span> is positive and it has a positive eigen vector(Perron-Frobenius Theorem)</p>
<p>How can I give an upper bound on <span class="math-container">$\rho(A)$</span> in terms of chromatic number of <span class="math-container">$G$</span>?</p>
<p>I want the bound to be proved in terms of chromatic number and in terms of number of vertices and edges of the graph <span class="math-container">$G$</span>.</p>
<p>How can I do it?</p>
<p><strong>Note:I checked Wilf's and Hoffmans bound.But I want to find some other bounds which involve using the Perrons Theorem and uses <span class="math-container">$n,m$</span></strong></p>
http://www.4124039.com/q/3306160Construction of a special sequence [duplicate]Kacdimahttp://www.4124039.com/users/1188482019-05-03T09:42:04Z2019-05-03T10:44:50Z
<div class="question-status question-originals-of-duplicate">
<p>This question is an exact duplicate of:</p>
<ul>
<li>
<a href="/questions/324167/a-special-sequence" dir="ltr">A special sequence</a>
</li>
</ul>
</div>
<p>I m looking for a sequence <span class="math-container">$(f_j)\in C^\infty(\Bbb{R})$</span> such that<br>
<span class="math-container">$$
\int^\infty_0\Big|4r\partial^2_r f_j(r)+4\partial_r f_j(r)+rf_j(r)\Big|^2dr\to 0,
$$</span> and </p>
<p><span class="math-container">$$\int_{\Bbb{R^+}}|f_j(r)|^2 dr=1\quad\forall j\in\Bbb{N}.$$</span></p>
<p>Could anybody help? or give some suggestions? Thanks in advance.</p>
http://www.4124039.com/q/330259-1Invariance of spectrum under conjugationuser136400http://www.4124039.com/users/1364002019-04-29T07:10:20Z2019-04-29T13:54:33Z
<p>Let <span class="math-container">$T$</span> be a self-adjoint invertible operator on <span class="math-container">$\mathcal{H}$</span> with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators <span class="math-container">$V$</span>( with continuous spectrum) other than unitary <span class="math-container">$VTV^{*}$</span> also have continuous spectrum?</p>
http://www.4124039.com/q/3140962The effect of random projections on matricesnevereverneverhttp://www.4124039.com/users/1230752018-10-29T20:16:37Z2019-04-28T02:01:25Z
<p>Let <span class="math-container">$A\in\mathbb{R}^{n\times n}$</span> be a given normal matrix, i.e. <span class="math-container">$A^TA=AA^T$</span>. Let <span class="math-container">$P_s\in\mathbb{R}^n$</span> be a random projection matrix to an <span class="math-container">$s$</span>-dimensional subspace in <span class="math-container">$\mathbb{R}^n$</span>. </p>
<p>Suppose <span class="math-container">$\frac{A+A^T} {2}$</span> is positive semidefinite with rank <span class="math-container">$l$</span>. Then can we say anything about the distribution of the eigenvalues of the following random matrix?
<span class="math-container">$$\frac{PA+A^TP}{2}$$</span>
Is it still positive semidefinite? Let <span class="math-container">$\mu_1,...,\mu_k$</span> be the non-zero eigenvalues of it where <span class="math-container">$k\leq s\wedge l$</span>. What is the expectation of <span class="math-container">$\prod_{i=1}^k\mu_i$</span>?</p>
http://www.4124039.com/q/3279073Oscillatory integralsAlihttp://www.4124039.com/users/504382019-04-12T16:37:07Z2019-04-19T23:22:25Z
<p>Consider the integrals</p>
<p><span class="math-container">$$I_n(\zeta,\epsilon)=\int_{-\zeta}^\zeta \left|(t-i\epsilon)^{-n}-(t+i\epsilon)^{-n}\right|\,dt$$</span></p>
<p>I would like to know the asymptotic behavior of <span class="math-container">$I_n(\zeta,\epsilon)$</span> for each fixed <span class="math-container">$\zeta>0$</span> as <span class="math-container">$\epsilon$</span> approaches zero, and hope that this will be independent of <span class="math-container">$\zeta$</span>. For example, when <span class="math-container">$n=1$</span>, it is easy to see that <span class="math-container">$I_1(\zeta,\epsilon)$</span> tends to <span class="math-container">$2\pi$</span> as <span class="math-container">$\epsilon$</span> tends to <span class="math-container">$0$</span> independent of <span class="math-container">$\zeta$</span>. Specifically, I would like to understand the cases where <span class="math-container">$n=\frac{1}{2},\frac{3}{2},\ldots$</span>, but the case of integers would also be interesting.
Note that the question is trivial without the absolute value in the integrand but I would like to see how much does the presence of the absolute value changes the asymptotic behavior.
Thanks,</p>
http://www.4124039.com/q/3282981Stable region of minimal hypersurfaces with finite Morse indexOnil90http://www.4124039.com/users/863412019-04-17T15:46:14Z2019-04-17T15:46:14Z
<p>In <a href="https://link.springer.com/content/pdf/10.1007/BF01394782.pdf" rel="nofollow noreferrer">this</a> <em>Inventiones Mathematicae</em> paper, Fischer-Colbrie proved the following result (Proposition 1):</p>
<p><strong>Proposition:</strong> Let <span class="math-container">$ M$</span> be a complete two-sided minimal surface in a three manifold <span class="math-container">$N$</span>. Then if <span class="math-container">$M$</span> has finite index, there exists a compact set <span class="math-container">$C \subseteq M$</span> such that <span class="math-container">$M\setminus C$</span> is stable and there exists a positive function <span class="math-container">$u$</span> on <span class="math-container">$M$</span> such that <span class="math-container">$L u = 0$</span> on <span class="math-container">$M\setminus C$</span>, where <span class="math-container">$L$</span> is the stability operator coming from the second variation of the area functional. </p>
<p>My question is if this statement is true in any dimension (assuming codimension <span class="math-container">$1$</span>). I'm reading the proof and it seems to me that the argument is independent from the dimension, but maybe I'm wrong. </p>
<p>Any help will be very appreciated! </p>
http://www.4124039.com/q/2998833Discrete spectrum of Schrodinger operatorDLINhttp://www.4124039.com/users/952962018-05-10T12:39:55Z2019-04-16T16:25:25Z
<p>Assume $\Omega$ is a non-compact region or manifold with dimension $\geq4$. Let $H=-\Delta+V$ be Schrodinger operator. Here $V$ is a (smooth)function. </p>
<p>I know that if $V\geq c>0$ or $V\to c>0$, then $0$ does not locate in the essential spectrum of $H$.s</p>
<p><strong>Q</strong> :
Is there any work to consider the negative case, i.e. $V>-c$, here $c>0$ is a constant, with what condition on $V$, we also have that $0$ locates in the discrete spectrum of $H$</p>
http://www.4124039.com/q/3279186Weyl law for (non-semiclassical) Schrodinger operatorMaxim Bravermanhttp://www.4124039.com/users/743072019-04-12T18:42:04Z2019-04-16T15:58:09Z
<p>The Weyl law for a semiclassical Schrodinger operator
<span class="math-container">$$ A_h\ := \ -h^2\Delta+V(x) $$</span>
on an <span class="math-container">$d$</span>-dimensional complete Riemannian manifold <span class="math-container">$M$</span>
says that the number <span class="math-container">$N(A_h,1)$</span> of eigenvalues of <span class="math-container">$A_h$</span> which are smaller than 1 has asymptotic behavior
<span class="math-container">$$
N(A_h,1)\ \sim \
\frac1{(2\pi h)^d}\ \text{Vol}\
{\Large(}(x,\xi)\in T^*M:\ |\xi|^2+V(x)\le 1{\Large)}, \quad h\to 0.\ \ \ (\ast)
$$</span>
I am interested in a non-semiclassical Schrodinger operator
<span class="math-container">$$ A\ := \ -\Delta+V(x).$$</span>
I believe that the number <span class="math-container">$N(A,\lambda)$</span> of eigenvalues of <span class="math-container">$A$</span> smaller than <span class="math-container">$\lambda$</span> has a similar asymptotic
<span class="math-container">$$ N(A,\lambda) \sim \ \frac1{(2\pi)^d} \text{Vol}\
{\Large(}(x,\xi)\in T^*M:\ |\xi|^2+V(x)\le \lambda{\Large)}, \qquad \lambda\to \infty.
\quad(\ast\ast)
$$</span>
This is, of course, true on a compact manifolds, due to the classical Weyl's law. It is also not difficult to verify (**) for operator <span class="math-container">$-\Delta+|x|^n$</span> on <span class="math-container">$\mathbb{R}^d$</span> (where <span class="math-container">$A_h$</span> and <span class="math-container">$A$</span> can be related by rescaling of <span class="math-container">$x$</span>). </p>
<p>So my question: is (**) true and, if it is true, where can I find it? </p>
<p>PS. Victor Ivrii in his review article <a href="https://arxiv.org/abs/1608.03963" rel="noreferrer">https://arxiv.org/abs/1608.03963</a> mentions (page 5) that, using Birman-Schwinger principle, one can obtain a Weyl law for <span class="math-container">$N(A,\lambda)$</span> from (*). But I don't see how it can be done. </p>
http://www.4124039.com/q/3281611Sturm-Liouville-like Eigenproblem¦Ðr8http://www.4124039.com/users/1216922019-04-16T00:06:10Z2019-04-16T00:06:10Z
<p>Consider the piecewise-deterministic Markov process on <span class="math-container">$\mathbf{R}$</span> which</p>
<ul>
<li>moves according to the vector field <span class="math-container">$\phi (x) = 1$</span>, </li>
<li>experiences events at rate <span class="math-container">$\lambda(x) = 1$</span>, and </li>
<li>at events, jumps according to the transition kernel</li>
</ul>
<p><span class="math-container">\begin{align}
Q (x \to dy) = \frac{\exp(- y - \exp(-y))}{\exp( - \exp (-x))} \, \mathbf{I} [y \leqslant x] dy
\end{align}</span></p>
<p>i.e. the location is resampled according to a Gumbel distribution, conditioned to be to the left of <span class="math-container">$x$</span>. </p>
<p>It can be shown that this process admits the Gumbel distribution as its stationary measure, that is, the measure with density <span class="math-container">$G(x) = \exp(- x - \exp(-x))$</span>. Moreover, its generator is given by</p>
<p><span class="math-container">\begin{align}
\mathcal{L} f (x) = f'(x) + \int_{-\infty}^x [f(y) - f(x)] Q (x \to dy).
\end{align}</span></p>
<p>It is worth remarking that the process is non-reversible, and hence its generator is not self-adjoint. </p>
<p>For reference, the generator's adjoint in <span class="math-container">$L^2 (dG)$</span> is given by</p>
<p><span class="math-container">\begin{align}
\mathcal{L}^* f (x) = - f'(x) + \exp (-x) \int_x^{\infty} [f(y) - f(x)] Q^* (x \to dy)
\end{align}</span></p>
<p>where</p>
<p><span class="math-container">\begin{align}
Q^* (x \to dy) = \exp(- (y - x)) \mathbf{I} [y \geqslant x] dy,
\end{align}</span></p>
<p>which can be derived by computing the time-reversal of the original process.</p>
<p>Now, I'm interested in studying the eigenfunctions of the generator, i.e. solutions to</p>
<p><span class="math-container">\begin{align}
\mathcal{L} f (x) = \lambda f (x)
\end{align}</span></p>
<p>After some calculations, I can simplify this equation to</p>
<p><span class="math-container">\begin{align}
f''(x) + (\exp ( -x) - 1 ) f'(x) = \lambda \left\{ \exp (-x) f(x) + f' (x) \right\}
\end{align}</span></p>
<p>which can be put into `quasi-Sturm-Liouville' form as</p>
<p><span class="math-container">\begin{align}
\frac{d}{dx} \left( G(x) f'(x) \right) &= \lambda G(x) \left\{ f' (x) + \exp (-x) f(x)\right\},
\end{align}</span></p>
<p>and even into the form</p>
<p><span class="math-container">\begin{align}
\frac{d}{dx} \left( G(x) f'(x) \right) &= \lambda \exp (-x) \frac{d}{dx} \left( \exp ( - \exp (-x)) f(x) \right),
\end{align}</span></p>
<p>which could conceivably be useful.</p>
<p>Anyways, although this all looks very reminiscent of Sturm-Liouville theory, it's not <em>quite</em> there, in as much as the term with the eigenvalue <span class="math-container">$\lambda$</span> don't just involve <span class="math-container">$f$</span>, but also its derivatives. </p>
<p><strong>My main questions are thus as follows:</strong></p>
<ol>
<li><p>Is there a name for this type of system? i.e. S-L-type structure, but where the <span class="math-container">$\lambda$</span>-term depends on the derivatives of <span class="math-container">$f$</span> as well. If so, I'd appreciate relevant references.</p></li>
<li><p>Are there any other techniques which might be of use in trying to solve this eigensystem? Ideally there will be solutions of the form </p></li>
</ol>
<p><span class="math-container">\begin{align}
f(x; \lambda_n) = \{ \text{polynomial of degree } n \} \times \text{fixed function},
\end{align}</span></p>
<p>but this might be a bit (/very) optimistic on my part. </p>
<p>A related consideration (which I'm in the process of doing the calculations for) is to carry out a spectral decomposition of <span class="math-container">$\mathcal{L} \mathcal{L}^*$</span> and <span class="math-container">$\mathcal{L}^* \mathcal{L}$</span>. Given that these will be self-adjoint, some of the theory might make life a little easier; on the other hand, both operators involve double integrals, so it may be time-consuming. I'll update with details once I've had a go of that. Any advice on that would also certainly be welcome.</p>
http://www.4124039.com/q/2699905Norm bounds on spectral variation and eigenvalue variationT. Amdeberhanhttp://www.4124039.com/users/661312017-05-17T02:02:58Z2019-04-15T07:16:09Z
<p>Let $A$ and $B$ be two matrices of eigenvalues $\lambda_i$ and $\mu_i$, respectively. </p>
<p>The <em>spectral variation</em> of $B$ w.r.t. $A$ and the <em>eigenvalue variation</em> of $B$ and $A$ are, respectively,
\begin{align} s_B(A)&=\max_i\min_j\vert\lambda_i-\mu_j\vert, \\
v(A,B)&=\min_{\pi}\max_i\vert\lambda_i-\mu_{\pi(i)}\vert;\end{align}
where in the latter the minimum is to be taken over all permutations $\pi$ of the indices.</p>
<blockquote>
<p><strong>Question 1.</strong> If $A$ and $B$ are Hermitian matrices, then for which norms is this true?
$$s_B(A)\leq\Vert A-B\Vert.$$</p>
<p><strong>Question 2.</strong> If $A$ and $B$ are normal matrices (more generally for fully symmetric operators), then for which norms is this true?
$$v(A,B)\leq\Vert A-B\Vert.$$</p>
</blockquote>
<p>I would appreciate any reference to the state-of-the-art in this matter.</p>
http://www.4124039.com/q/3269713Is the ring of $p$-adic integers extremally disconnected?Rick Sternbachhttp://www.4124039.com/users/1285402019-04-02T13:53:43Z2019-04-14T21:17:43Z
<p>We call a topological space <span class="math-container">$X$</span> extremally disconnected if the closures of its open sets remain open. Obviously, Hausdorff extremally disconnected spaces are totally disconnected in the sense that their connected components are singletons.</p>
<p>Suppose <span class="math-container">$\mathbb{Z}_p = \varprojlim_{n \ge 1} \mathbb{Z}/p^n\mathbb{Z}$</span> is the ring of <span class="math-container">$p$</span>-adic integers, viewed as a closed subspace of the compact Hausdorff space <span class="math-container">$\prod_{n \ge 1} \mathbb{Z}/p^n\mathbb{Z}$</span>, where the latter is equipped with the product topology and each factor <span class="math-container">$\mathbb{Z}/p^n\mathbb{Z}$</span> is discret. Then it is well-known that <span class="math-container">$\mathbb{Z}_p$</span> is totally disconnected. But I can't answer the seemingly naive question: is <span class="math-container">$\mathbb{Z}_p$</span> extremally disconnected?</p>
<p>Following the traditions of operator algebraists, we call a compact Hausdorff extremally disconnected space a stonean space. And we call a stonean space hyperstonean if it admits enough normal measures---a technical condition which characterises hyperstonean spaces exactly as the spectrum of abelian von Neumann algebras. A series of possibly less naive questions are: if <span class="math-container">$\mathbb{Z}_p$</span> is stonean (which is the same as its extremal disconnectedness), is it hyperstonean? And if it is hyperstonean, can we associate some natural Hilbert space (like <span class="math-container">$L^2(\mathbb{Z}_p, \mu)$</span>, where <span class="math-container">$\mu$</span> is the normalized Haar measure on <span class="math-container">$\mathbb{Z}_p$</span> viewed as a compact abelian topological group), and an abelian von Neumann algebra (hopefully the algebra <span class="math-container">$C(\mathbb{Z}_p)$</span> of complex-valued continuous functions on <span class="math-container">$\mathbb{Z}_p$</span>) acting on this Hilbert space, such that <span class="math-container">$\mathbb{Z}_p$</span> is exactly the spectrum of this latter von Neumann algebra?</p>
<p>Finally, a perhaps stupid and somewhat less well-formulated question: do we have some "nice" examples of stonean, even hyperstonean spaces, that are infinite, and are not constructed as the spectrums of their associated operator algebras (of course they do appear as such spectrums, I mean one describe the relevant topologies without referring first to the algebra of functions on it)?</p>
<p>I ask these questions because as far as I know, unlike the widely spread notion of totally disconnected spaces (or <span class="math-container">$0$</span>-dimensional spaces), (hyper)stonean spaces seems to be only of a minor interest to operator algebraists as spectrums of abelian von Neumann algebras (except Gleason's theorem characterising them as the projective object in the category of compact Hausdorff spaces). And the only way of producing nontrivial stonean spaces that I am aware of is taking the spectrum of some algebras. If <span class="math-container">$\mathbb{Z}_p$</span> does turn out to be stonean, it would be a cute concrete example in my humble opinion.</p>
http://www.4124039.com/q/2030283Proof of eigenvalue stability inequality via Courant-Fischer min-max theoremTahahttp://www.4124039.com/users/344452015-04-15T19:24:45Z2019-04-12T19:56:02Z
<p>Dr. Tao in <a href="https://terrytao.wordpress.com/2010/01/12/254a-notes-3a-eigenvalues-and-sums-of-hermitian-matrices/" rel="nofollow">his notes on eigenvalue inequalities uses Courant-Fischer min-max theorem to prove the eigenvalue stability inequality</a>. Specifically, I am looking for proof of Eq. (13) where Dr. Tao states as an immediate result of Eq. (6) and (10). But the problem is that the min-max function is not convex. I have read Stewart & Sun's book on <em>Matrix Perturbation Theory</em>, but it seems that they have felt that it is obvious too. </p>
<p>Can someone provide more details on how to derive Eq. (13)?</p>
http://www.4124039.com/q/3278172Smallest singular value distributionDominikhttp://www.4124039.com/users/1337522019-04-11T21:03:11Z2019-04-11T21:03:11Z
<p>Let <span class="math-container">$G_\mathbb{R}\in\mathbb{R}^{n\times n}$</span> and <span class="math-container">$G_\mathbb{C}\in\mathbb{C}^{n\times n}$</span> denote the real and complex Ginibre random matrices, i.e. random matrices with independent real/complex Gaussian entries of zero mean and variance <span class="math-container">$\mathbb{E}\lvert(G_K)_{ij}\rvert^2=n^{-1}$</span> for <span class="math-container">$K=\mathbb{R},\mathbb{C}$</span>. Denote the singular values of the matrix <span class="math-container">$X$</span> by <span class="math-container">$\sigma_1(X)\ge\dots\ge \sigma_n(X)$</span>. It follows from the work of Edelman that
<span class="math-container">\begin{equation}\mathbb{P}\Big( \sigma_n(G_K)\le\frac{x}{n}\Big) = \begin{cases} 1-e^{-x^2/2-x}+o(1),&K=\mathbb{R}\\
1-e^{-x^2},&K=\mathbb{C}.
\end{cases}\tag{*}\label{sing val}\end{equation}</span>
In particular, the smallest singular value is of order <span class="math-container">$n^{-1}$</span>.</p>
<p>Now I am wondering about additive perturbations to <span class="math-container">$G_K$</span>, say <span class="math-container">$G_K+\lambda I$</span>, where <span class="math-container">$\lambda$</span> is some real (or even complex) parameter and <span class="math-container">$I$</span> is the identity matrix. In general the singular values of <span class="math-container">$G_K$</span> give little information about the singular values of <span class="math-container">$G_K+\lambda I$</span>. The most one can hope for is bounds like
<span class="math-container">$$\sigma_n(G_K+\lambda I)\ge \lvert\lambda\rvert-\sigma_1(G_K).$$</span>
It is known that <span class="math-container">$\sigma_1(G_K)$</span> is Tracy-Widom distributed around <span class="math-container">$2$</span>, so in particular, <span class="math-container">$\sigma_n(G_k+\lambda I)$</span> is bounded away from <span class="math-container">$0$</span> as long as <span class="math-container">$\lvert \lambda\rvert>2$</span>. </p>
<p><strong>Question:</strong> Is the analogue of <span class="math-container">$\eqref{sing val}$</span>, i.e. the distribution of <span class="math-container">$\sigma_n(G_K+\lambda I)$</span> known for <span class="math-container">$G_K+\lambda I$</span>? If exact formulae are not available, I would be interested in the average scaling of <span class="math-container">$\sigma_n(G_K+\lambda I)$</span>. I guess there should be some phase transition of the type
<span class="math-container">$$\mathbb E \sigma_n(G_K+\lambda I)\sim\begin{cases}n^{-1},&\lvert\lambda\rvert<c\\
n^{-?}, &\lvert\lambda\rvert=c,\\
1, &\lvert\lambda\rvert>c.\end{cases}$$</span>
I think <span class="math-container">$c=1$</span>, but am unsure about the critical exponent. </p>
http://www.4124039.com/q/32733011Smoothness of finite-dimensional functional calculusMizarhttp://www.4124039.com/users/369522019-04-06T17:05:11Z2019-04-06T20:36:05Z
<p>Assume that <span class="math-container">$f:\mathbb R\to\mathbb R$</span> is continuous.
Given a real symmetric matrix <span class="math-container">$A\in\text{Sym}(n)$</span>, we can define <span class="math-container">$f(A)$</span> by applying <span class="math-container">$f$</span> to its spectrum. More explicitly,
<span class="math-container">$$ f(A):=\sum f(\lambda)P_\lambda,\qquad A=\sum\lambda P_\lambda. $$</span>
Here both sums are finite, and the second one is the decomposition of <span class="math-container">$A$</span> as a linear combination of orthogonal projections (<span class="math-container">$P_\lambda$</span> is the projection onto the eigenspace for the eigenvalue <span class="math-container">$\lambda$</span>, so that <span class="math-container">$P_\lambda P_{\lambda'}=0$</span>). Such decomposition exists and is unique by the spectral theorem.</p>
<p>I guess it is well known that <span class="math-container">$f:\text{Sym}(n)\to\text{Sym}(n)$</span> is continuous.</p>
<blockquote>
<p>Assuming <span class="math-container">$f\in C^\infty(\mathbb R)$</span>, is the induced map <span class="math-container">$f:\text{Sym}(n)\to\text{Sym}(n)$</span> also smooth?</p>
</blockquote>
<p>I think I can show that it is (Fréchet) differentiable everywhere, but I am wondering whether it is always <span class="math-container">$C^1$</span> or even <span class="math-container">$C^\infty$</span>.</p>
http://www.4124039.com/q/3263923Quasinilpotent vectors of Newton potential vanishBowenhttp://www.4124039.com/users/1343332019-03-26T12:24:17Z2019-04-01T08:42:27Z
<p>Suppose <span class="math-container">$\Omega$</span> is a smooth bounded domain in <span class="math-container">$\mathbb{R}^3$</span>. Consider the Newton potential
<span class="math-container">\begin{equation}
T [\phi](x) = \int_{\Omega} \frac{1}{|x-y|} \phi(y)dy.
\end{equation}</span>
It is well know that <span class="math-container">$T$</span> is a bounded linear operator from <span class="math-container">$L^2(\Omega)$</span> to <span class="math-container">$H^2(\Omega)$</span>. Hence it is a self adjoint compact operator defined on <span class="math-container">$L^2(\Omega)$</span>. Suppose that it has the following spectral decomposition:
<span class="math-container">$$T \phi = \sum^\infty_{j = 1}\lambda_j (\phi,e_j) e_j,$$</span>
where <span class="math-container">$(\lambda_j,\phi_j)$</span> is the eigenpair counting multiplicity. And we can see <span class="math-container">$ker T = \{0\}$</span> from the following observation: <span class="math-container">$\Delta T[\phi] = C\phi$</span> on <span class="math-container">$\Omega$</span> for some positive constant <span class="math-container">$C$</span>. </p>
<p>We say that a vector <span class="math-container">$q$</span> in <span class="math-container">$L^2(\Omega)$</span> is a quasinilpotent vector if
<span class="math-container">$$
\lim_{n \to \infty}||T^n q||^{\frac{1}{n}} = 0.
$$</span>
Then from above spectral decomposition and fact that <span class="math-container">$\lambda_j > 0$</span>, we can claim all the quasinilpotent vectors of <span class="math-container">$T$</span> vanish. Indeed, if <span class="math-container">$\phi$</span> is a quasinilpotent vector, then
<span class="math-container">$$ \lim_{n \to \infty}|(e_j,T^n \phi)|^{\frac{1}{n}} = \lambda_j |(e_j,\phi)|^{1/n} = 0 ,$$</span>
which gives us <span class="math-container">$(e_j,\phi)$</span> vanishes for all <span class="math-container">$j$</span>. </p>
<p>I would like to prove the same result (all the quasinilpotent vectors vanish) for the following operator,
<span class="math-container">$$ T_k[\phi] = \int_{\Omega} \frac{e^{ik|x-y|}}{|x-y|}\phi(y)dy,$$</span>
which is a also a compact operator on <span class="math-container">$L^2(\Omega)$</span>. But we may not expect the above arguments work in our case since the spectral structure of <span class="math-container">$T_k$</span> is not clear. Perhaps we need turn to elliptic PDE theory for help.</p>
<p>Thank you very much in advance for any insight or suggestions.</p>
http://www.4124039.com/q/3268642Existence of a fixed point for this operatorCAPMhttp://www.4124039.com/users/972322019-04-01T06:43:48Z2019-04-01T06:43:48Z
<p>I'm looking for some mathematical results that I might be able to apply to see if an operator I'm considering has a fixed point.</p>
<p>In particular consider,
<span class="math-container">$$ Ag(x) = \Big\{ \xi(x) + K(g(x))^\frac{1}{\theta} \Big\}^\theta $$</span>
where <span class="math-container">$g$</span> is some <span class="math-container">$L^1(X)$</span> or <span class="math-container">$C^1(X)$</span> (if easier) function, <span class="math-container">$X$</span> is a compact set, <span class="math-container">$K$</span> is a linear operator and <span class="math-container">$\xi$</span> is compact valued and positive. I know for a fact that if the spectral radius condition <span class="math-container">$r(K)^\frac{1}{\theta}<1$</span> holds then <span class="math-container">$K$</span> will have a fixed point and so if <span class="math-container">$\xi(x) = C$</span> were a constant then <span class="math-container">$A$</span> would have a fixed point.</p>
<p>I'm wondering how to extend it to the case where <span class="math-container">$\xi(x)$</span> is compact valued. </p>
<p>One idea I had would be to note that by compactness <span class="math-container">$\xi(x)$</span> has an upper and lower bound and for these 'upper' and 'lower' versions of <span class="math-container">$A$</span> we just consider <span class="math-container">$$A_0g(x) = \Big\{ C + Kg(x)^\frac{1}{\theta} \Big\}^\theta \leq \Big\{ \xi(x) + Kg(x)^\frac{1}{\theta} \Big\}^\theta \leq \Big\{ C' + Kg(x)^\frac{1}{\theta} \Big\}^\theta = A_1g(x)$$</span></p>
<p>Since <span class="math-container">$A_1$</span> and <span class="math-container">$A_0$</span> have fixed points is there some kind of result that will get <span class="math-container">$A$</span> to have a fixed point? I tried taking an interpolation argument by defining for each <span class="math-container">$x \in X$</span> a constant <span class="math-container">$\xi(x) = \xi_x$</span> such that <span class="math-container">$A$</span> has a fixed point <span class="math-container">$g_x$</span> and then defining <span class="math-container">$g^*(x) = g_x (x) $</span>. The problem would then reduce to showing that <span class="math-container">$g^* \in C^1(X)$</span>. </p>
<p>Any help would be appreciated.</p>
http://www.4124039.com/q/3264071The Morse Index of a $T$- periodic geodesics is a integer number?Marcelo Nogueirahttp://www.4124039.com/users/1370682019-03-26T16:13:34Z2019-03-26T16:13:34Z
<p>It is well known that compact Riemannian manifolds <span class="math-container">$(M, g)$</span> with
periodic geodesic flows have ( Besse Book) exceptional spectral properties: the spectrum
of <span class="math-container">$ \sqrt{ - \Delta}$</span>, the square root of the Laplacian, concentrates along the arithmetic
progression [<span class="math-container">$(\frac{ 2 \pi}{T}$</span>) <span class="math-container">$(k + \beta)$</span>: <span class="math-container">$k=1, 2, ...$</span>] with <span class="math-container">$T$</span> the (minimal) period of
the geodesic flow <span class="math-container">$G^{t}$</span> and with ; <span class="math-container">$\beta$</span> the common Morse index of the <span class="math-container">$T$</span>-periodic
geodesics. </p>
<p>My question is: <span class="math-container">$\beta$</span> is a integer number if <span class="math-container">$T = \pi$</span>, for example ? </p>
http://www.4124039.com/q/3241332Non-isolated ground state of a Schrödinger operatorJochen Glueckhttp://www.4124039.com/users/1029462019-02-25T22:49:25Z2019-03-26T03:18:34Z
<p><strong>Question.</strong> Does there exist a dimension <span class="math-container">$d \in \mathbb{N}$</span> and a measurable function <span class="math-container">$V: \mathbb{R}^d \to [0,\infty)$</span> such that the smallest spectral value <span class="math-container">$\lambda$</span> of the Schrödinger operator <span class="math-container">$-\Delta + V$</span> on <span class="math-container">$L^2(\mathbb{R}^d)$</span> is an eigenvalue, but not an isolated point of the spectrum?</p>
<p>I would expect this to be known, but I could not come up with an example (neither myself nor by browsing some manuscripts about Schrödinger operators).</p>
http://www.4124039.com/q/3236123Real part of eigenvalues and LaplacianYizhao Sunhttp://www.4124039.com/users/1360332019-02-20T00:36:55Z2019-03-22T04:56:41Z
<p>I am working on imaging and I am a bit puzzled by the behaviour of this matrix: </p>
<p><span class="math-container">$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 1 & 0 & 0 & -1 \\
2 & -2 & 0 & 0 & 0 & 0 \\
-2 & 4 & -2 & 0 & 0 & 0 \\
0 & -2 & 2 & 0 & 0 & 0 \\
\end{array}
\right)$$</span></p>
<p>My matrix <span class="math-container">$A$</span> is a <span class="math-container">$4x4$</span> block matrix with an upper block <span class="math-container">$A_{11}:= \operatorname{diag}(1,0,1)$</span>, a second <span class="math-container">$A_{12}= -id$</span>, a lower block which is the graph Laplacian <span class="math-container">$-\Delta$</span> and then a block of zeros.</p>
<p>It is known that the lowest eigenvalue of the graph Laplacian is zero.</p>
<p>Now my matrix has all eigenvalues on the right hand side of the complex plane (non-negative real part) and the one with smallest real part has real part zero.</p>
<p>However, if I consider instead of the graph Laplacian in the <span class="math-container">$A_{21}$</span> block the matrix <span class="math-container">$-\Delta+id$</span> then this block is bounded away from zero and </p>
<p><span class="math-container">$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 1 & 0 & 0 & -1 \\
2+1 & -2 & 0 & 0 & 0 & 0 \\
-2 & 4+1 & -2 & 0 & 0 & 0 \\
0 & -2 & 2+1 & 0 & 0 & 0 \\
\end{array}
\right)$$</span></p>
<p>has only eigenvalues with strictly positive real part. </p>
<p>I ask: Can anybody explain the relationship between the lower left corner of my matrix <span class="math-container">$A$</span> having spectrum bounded away from zero and all eigenvalues of <span class="math-container">$A$</span> being strictly contained in the right half plane?</p>
<p>How does <span class="math-container">$\lambda_{\text{min}}(A_{21})$</span> relate to <span class="math-container">$\operatorname{min}\Re(\sigma(A))$</span>?</p>
<p>EDIT: I was thinking that some Block matrix identities may be useful <a href="http://djalil.chafai.net/blog/2012/10/14/determinant-of-block-matrices/" rel="nofollow noreferrer">http://djalil.chafai.net/blog/2012/10/14/determinant-of-block-matrices/</a></p>
<p>by I do not quite get it together.</p>
http://www.4124039.com/q/3254843Mixing time and spectral gap for a special stochastic matrixHao Yuanhttp://www.4124039.com/users/1004822019-03-15T00:28:07Z2019-03-15T15:23:06Z
<p>Conisder the following dimension stochastic matrix,
<span class="math-container">\begin{bmatrix}
p & q & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 & 0 \\
\end{bmatrix}</span>
with <span class="math-container">$p,q>0$</span> and <span class="math-container">$p+q=1$</span>. </p>
<p>To control mixing time, I am interested in its spectral gap. Let <span class="math-container">$n$</span> be the number of rows (e.g., <span class="math-container">$n=5$</span> in the example above). The characteristic polynomial is
<span class="math-container">$$x^n - px^{n-1} - q.$$</span>
How to argue that the spectral gap decay polynomially in terms of <span class="math-container">$n$</span> (rather than of exponentially)?</p>
http://www.4124039.com/q/3254054The exceptional eigenvalues and Weyl's law in level aspectQinghua Pihttp://www.4124039.com/users/456912019-03-14T01:16:01Z2019-03-14T11:42:49Z
<p>The Weyl law for Maass cusp forms for <span class="math-container">$SL_2(\mathbb Z)$</span> was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of the version of Weyl's law for maass wave forms for congrunce subgroup is
<span class="math-container">$$
\sum_{t_j\in \mathbb R\atop{|t_j|<T}}1=\frac{\mathrm{Vol}(\Gamma_0(N)\backslash \mathbb{H})}{2\pi}T^2+O\left(T\log T\right).
$$</span>
Here <span class="math-container">$\{\pm t_j\}$</span> is the set of spectral parameters of Maass form and <span class="math-container">$\lambda_j=\frac{1}{4}+t_j^2$</span> is the eigenvalue of the Laplacian operator.</p>
<p>For <span class="math-container">$N$</span> large, the Selberg eigenvalue conjecture has not been proved. Some result (Xianjin Li) indicated that the multiplicity of exceptional eigenvalue (if it exists) can be arbituary large and is bounded by a constant depending on <span class="math-container">$N$</span>. </p>
<p>Is it possible to obtain an upper bound of the exceptional eigenvalues such as
<span class="math-container">$$
\sum_{t_j\in i\mathbb R\atop{0<|t_j|<1/2}}1\ll \frac{{\mathrm{Vol}(\Gamma_0(N)\backslash\mathbb H)}}{\log N},\quad N\rightarrow\infty
$$</span>
or equivalently, can we get rid of the contribution of exceptional eigenvalues in the proof of Weyl's law to obtain a version in level aspect,
<span class="math-container">$$
\sum_{t_j\in \mathbb R\atop{|t_j|<T}}1=c_T\mathrm{vol}(\Gamma_0(N)\backslash \mathbb H) +O\left(\frac{\mathrm{Vol}(\Gamma_0(N)\backslash\mathbb H)}{\log N}\right)?$$</span></p>
http://www.4124039.com/q/3243982Compact operators on Banach spaces and their spectrasharpehttp://www.4124039.com/users/684632019-03-01T09:10:15Z2019-03-11T17:13:57Z
<p>I have a question about compact operators on Banach spaces.</p>
<p>Let <span class="math-container">$B$</span> be a real Banach space and <span class="math-container">$L$</span> a closed linear operator on <span class="math-container">$B$</span>.
We assume that <span class="math-container">$L$</span> generates a contraction semigroup <span class="math-container">$\{T_t\}_{t>0}$</span> on <span class="math-container">$B$</span> . </p>
<p><strong>If <span class="math-container">$B$</span> is a Hilbert space</strong> and <span class="math-container">$L$</span> is self-adjoint, the following assertions are equivalent:</p>
<p>(1) The spectrum of <span class="math-container">$L$</span> is discrete (the essential spectrum <span class="math-container">$\sigma_{ess}(L)=\emptyset$</span>).</p>
<p>(2) <span class="math-container">$T_t$</span> is compact for any <span class="math-container">$t>0$</span>.</p>
<p>(3) <span class="math-container">$T_t$</span> is compact for some <span class="math-container">$t>0$</span>.</p>
<p>(4) <span class="math-container">$R_{\lambda}:=(\lambda-L)^{-1}$</span> is compact for any <span class="math-container">$\lambda \in \rho(L)$</span>.</p>
<p>(5) <span class="math-container">$R_{\lambda}$</span> is compact for some <span class="math-container">$\lambda \in \rho(L)$</span>.</p>
<p>Here, <span class="math-container">$\rho(L)$</span> is the resolvent set of <span class="math-container">$L$</span>.</p>
<p>Even if <span class="math-container">$B$</span> is not a Hilbert space, (2)<span class="math-container">$\Rightarrow$</span>(4), (4)<span class="math-container">$\Leftrightarrow$</span>(5), (5)<span class="math-container">$\Rightarrow$</span>(1).</p>
<p><strong>My question</strong></p>
<p>In what follows, we further assume that <span class="math-container">$\{T_t\}_{t>0}$</span> is strongly continuous and <span class="math-container">$B$</span> is a <span class="math-container">$L^1$</span> space on a measure space. </p>
<p>Does (1)<span class="math-container">$\Rightarrow$</span>(5) hold? or </p>
<p><strong>Under what conditions, does (1)<span class="math-container">$\Rightarrow$</span>(5) hold?</strong></p>
<p>By the way, I am particularly interested in situations where <span class="math-container">$\{T_t\}_{t>0}$</span> is generated by a <strong>symmetric</strong> Markov process on a locally compact metric measure space <span class="math-container">$(X,\mu)$</span>. In this case, for each <span class="math-container">$1\le p <\infty$</span>, <span class="math-container">$\{T_t\}_{t>0}$</span> is extended to a strongly continuous contraction semigroup <span class="math-container">$\{T_t^p\}_{t>0}$</span> on <span class="math-container">$L^{p}(X,\mu)$</span> and it holds that <span class="math-container">$T_t^p f=T_tf$</span> for any <span class="math-container">$t>0$</span> and <span class="math-container">$f \in L^{1}(X,\mu) \cap L^{p}(X,\mu)$</span>.</p>
http://www.4124039.com/q/3246031One question about Schrodinger Semigroups-(B. Simon)DLINhttp://www.4124039.com/users/952962019-03-04T13:42:16Z2019-03-10T14:03:27Z
<p>This question comes from the paper: B. Simon, <a href="https://projecteuclid.org/euclid.bams/1183549767" rel="nofollow noreferrer">Schrodinger Semigroups</a>, Bull. A.M.S., (1982) Vol. 7 (3). </p>
<p>On the <strong>Theorem C.3.4</strong>(subsolution estimate) of the paper, it says that: Let <span class="math-container">$Hu=Eu$</span> and <span class="math-container">$u\in L^2$</span>, where <span class="math-container">$H=-\Delta+V$</span> for some bounded continuous function <span class="math-container">$V$</span>, if <span class="math-container">$E$</span> is in discrete spectrum of <span class="math-container">$H$</span>. Then for some <span class="math-container">$C,~\delta>0$</span>
<span class="math-container">$$|u(x)|\leq Ce^{-\delta|x|}.$$</span></p>
<p><strong>Q</strong> </p>
<ul>
<li><p>How to get constant <span class="math-container">$C$</span>, does it depend <span class="math-container">$\delta$</span>? </p></li>
<li><p>Is there any estimate about <span class="math-container">$C$</span>, especially without the condition on the compactness of <span class="math-container">$supp(E-V)_+$</span>?</p></li>
</ul>
<p><em>PS</em>:
I can follow how to get <span class="math-container">$\delta$</span>. I can understand that <span class="math-container">$e^{\delta |x|}u\in L^\infty$</span>. I do not know how to get the constant <span class="math-container">$C$</span>, as the equation is linear, if we multiply any positive number <span class="math-container">$k$</span> with <span class="math-container">$u$</span>, it is also a solution for the equation, where I can not follow.
And I think <span class="math-container">$C$</span> depends on the <span class="math-container">$L^2$</span>-norm of <span class="math-container">$u$</span>.</p>
http://www.4124039.com/q/3226591Spectrum of the Magnetic Stark Hamiltonians $H(\mu,\epsilon)$Kacdimahttp://www.4124039.com/users/1188482019-02-07T10:38:00Z2019-03-10T12:03:00Z
<p>I am looking for a document where I can find a proof the spectrum of the of the Magnetic Stark
Hamiltonians <span class="math-container">$H(\mu,\epsilon)=\big(D_x-\mu y)^2+D^2_y+\epsilon x+V(x,y)$</span> cited on the article below for <span class="math-container">$\epsilon\not=0$</span> see equation <span class="math-container">$(1.2)$</span> in</p>
<p><a href="http://www.hrpub.org/download/20140105/MS6-13401691.pdf" rel="nofollow noreferrer">http://www.hrpub.org/download/20140105/MS6-13401691.pdf</a></p>
<p>Thanks</p>
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