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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
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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
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2 votes
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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
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
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1 vote
0 answers
127 views

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-...
lasik43's user avatar
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154 views

One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
Alphabeta's user avatar
1 vote
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576 views

Perturbation of Laplacian via Kato-Rellich theorem

Let's consider a potential $V(x)\in L^3(\mathbb{R}^3)$. I want to know if the following Hamiltonian $$-\Delta+V(x)$$ is self-adjoint on $H^2(\mathbb{R}^3)$. My idea is to use Kato-Rellich theorem; ...
Sam's user avatar
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Equality between operators, on dense subspace, from a quadratic form point of view

Let $L \ge 1$ and consider a finite box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$. The set of functions: $$\psi_{p}(x) = \frac{1}{L^{d/2}}e^{i\langle p,x\rangle} \quad p\in \frac{2\pi}{L}\mathbb{Z}^{...
MathMath's user avatar
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Question about a step in the proof of the min-max principle

I honestly do not think this is a hard question, maybe it is even obvious but I tried MSE and had no success so far, so I am reproducing the question Question about the proof of the min-max principle ...
MathMath's user avatar
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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 ...
user271621's user avatar
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237 views

Spectrum of a Hamiltonian on the real line

Consider the following linear (Hamiltonian) operator on functions on the real line $\mathbb{R}$ $$H\psi(x)=-\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x).$$ Assume that $V$ is a smooth function and $V(x)\to +\...
asv's user avatar
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