# Questions tagged [sp.spectral-theory]

Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices

**2**

votes

**0**answers

27 views

### Existence of a fixed point for this operator

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.
In particular consider,
$$ Ag(x) = \Big\{ \xi(x) + K(g(x))^\frac{1}{\...

**1**

vote

**0**answers

34 views

### The Morse Index of a $T$- periodic geodesics is a integer number?

It is well known that compact Riemannian manifolds $(M, g)$ with
periodic geodesic flows have ( Besse Book) exceptional spectral properties: the spectrum
of $ \sqrt{ - \Delta}$, the square root of ...

**3**

votes

**1**answer

121 views

### Quasinilpotent vectors of Newton potential vanish

Suppose $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$. Consider the Newton potential
\begin{equation}
T [\phi](x) = \int_{\Omega} \frac{1}{|x-y|} \phi(y)dy.
\end{equation}
It is well know ...

**3**

votes

**1**answer

93 views

### Mixing time and spectral gap for a special stochastic matrix

Conisder the following dimension stochastic matrix,
\begin{bmatrix}
p & q & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 &...

**4**

votes

**1**answer

134 views

### The exceptional eigenvalues and Weyl's law in level aspect

The Weyl law for Maass cusp forms for $SL_2(\mathbb Z)$ was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of ...

**4**

votes

**0**answers

58 views

### Perturbation theory compact operator

Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$
$\Vert Kx-\lambda x \Vert \le \varepsilon.$
It is well-known ...

**2**

votes

**0**answers

64 views

### Off-diagonal estimates for Poisson kernels on manifolds

Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...

**1**

vote

**0**answers

162 views

### One question about Schrodinger Semigroups-(B. Simon)

This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).
On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...

**2**

votes

**1**answer

321 views

### Compact operators on Banach spaces and their spectra

I have a question about compact operators on Banach spaces.
Let $B$ be a real Banach space and $L$ a closed linear operator on $B$.
We assume that $L$ generates a contraction semigroup $\{T_t\}_{t>...

**0**

votes

**0**answers

118 views

### A special sequence

I m looking for a sequence $(f_j)\in C^\infty(\Bbb{R})$ such that
$$
\int^\infty_0\Big|\partial^2_r f_j+\frac{1}{r}\partial_r f_j+r^2f_j\Big|^2rdr\to 0,
$$ and
$$\int_{\Bbb{R^+}}|f_j(r)|^2 rdr=1\...

**2**

votes

**1**answer

132 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 Schr?dinger operator $-\...

**6**

votes

**1**answer

158 views

### Eigenvalue estimates for operator perturbations

I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading.
What was behind ...

**5**

votes

**1**answer

163 views

### Perturbing a normal matrix

Let $N$ be a normal matrix.
Now I consider a perturbation of the matrix by another matrix $A.$
The perturbed matrix shall be called $M=N+A.$
Now assume there is a normalized vector $u$ such that $\...

**4**

votes

**2**answers

81 views

### Stable matrices and their spectra

I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices.
A matrix in our terminology is called stable if the real part of the eigenvalues is strictly ...

**3**

votes

**1**answer

132 views

### Real part of eigenvalues and Laplacian

I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...