PhyslibAlpha.AlgebraicFramework.HilbertSpace.Unbounded.Stone

For any real number $r$, the function $t \mapsto e^{itr}$ (where $i$ is the imaginary unit) has a derivative at $t = 0$ equal to $ir$.

For any real number $r$, the limit of the difference quotient of the function $t \mapsto e^{itr}$ at $t = 0$ is $ir$. That is, $$\lim_{t \to 0} \frac{e^{itr} - 1}{t} = ir$$ where $i$ is the imaginary unit.

For any real numbers $t$ and $r$ such that $|tr| \le 1$, the complex absolute value of $e^{itr} - 1$ is bounded by $2|tr|$, i.e., $|e^{itr} - 1| \le 2|tr|$.

For any real numbers $t$ and $r$ such that $t \neq 0$ and $|t| \leq 1$, the norm of the difference quotient of the function $e^{itr}$ relative to $t=0$ is bounded by $2|r|$. That is, $$\left\| \frac{e^{itr} - 1}{t} \right\| \leq 2|r|,$$ where $e^{itr}$ is the complex-valued exponential multiplier $x \mapsto e^{itx}$ evaluated at $x = r$.

For any real numbers $t$ and $r$ such that $t \neq 0$ and $|t| \leq 1$, the norm of the difference between the difference quotient of the exponential multiplier $e^{itr}$ and its derivative at zero is bounded by $3|r|$: $$\left| \frac{e^{itr} - 1}{t} - ir \right| \leq 3|r|$$ where $i$ is the imaginary unit.

For real parameters $t$ and $r$, this is the complex-valued function defined as the difference quotient $\frac{e^{itr} - 1}{t}$, where $i$ is the imaginary unit.

For any fixed real number $t$, the function $r \mapsto \frac{e^{itr} - 1}{t}$ is measurable, where $i$ is the imaginary unit.

For any real number $r$, the limit as $t$ approaches $0$ (with $t \neq 0$) of the difference quotient $\frac{e^{itr} - 1}{t}$ is equal to $ir$, where $i$ is the imaginary unit.

For any real number $t$, the function mapping $r \in \mathbb{R}$ to $\frac{e^{itr} - 1}{t} - ir$ is measurable, where $i$ is the imaginary unit.

Let $t$ and $r$ be real numbers. If $t \neq 0$ and $|t| \le 1$, then the norm of the difference between the difference quotient $\frac{e^{itr} - 1}{t}$ and the complex value $ir$ is bounded by $3|r|$: $$\left| \frac{e^{itr} - 1}{t} - ir \right| \le 3|r|,$$ where $i$ denotes the imaginary unit.

Let $\mu$ be a complex-valued vector measure on $\mathbb{R}$ such that the identity function $r \mapsto r$ is integrable with respect to $\mu$. Then, as $t \in \mathbb{R}$ tends to $0$ (with $t \neq 0$), the vector integral $$\int_{\mathbb{R}} \left( \frac{e^{itr} - 1}{t} - ir \right) d\mu(r)$$ tends to $0$, where $i$ denotes the imaginary unit.

Let $H$ be a Hilbert space and $\mu_S$ be a spectral measure on $\mathbb{R}$. Let $A$ be the operator defined by the maximal spectral integral $\int_{\mathbb{R}} r \, d\mu_S(r)$. For any vectors $x$ in the domain of $A$ and $y \in H$, if the identity function $r \mapsto r$ is integrable with respect to the scalar measure $\mu_{x,y}$ (defined by $\mu_{x,y}(S) = \langle y, \mu_S(S) x \rangle$), then the inner product of $y$ and $Ax$ is equal to the complex weak integral of the identity function: $$\langle y, Ax \rangle = \int_{\mathbb{R}} r \, d\mu_{x,y}(r).$$

Let $H$ be a Hilbert space and $\mu_S$ be a spectral measure on $\mathbb{R}$ with values in the operators on $H$. Let $x, y \in H$ be vectors such that the identity function $r \mapsto r$ is integrable with respect to the scalar measure $\mu_{x,y}$ (defined by $\mu_{x,y}(S) = \langle y, \mu_S(S) x \rangle$). Let $e^{itT} = \int_{\mathbb{R}} e^{itr} \, d\mu_S(r)$ be the unitary operator associated with the spectral measure for a real parameter $t$. Then, as $t \to 0$ (for $t \neq 0$), the difference quotient of the matrix element $\langle y, e^{itT} x \rangle$ converges to the complex weak integral of $ir$: $$\lim_{t \to 0} \frac{\langle y, e^{itT} x \rangle - \langle y, x \rangle}{t} = \int_{\mathbb{R}} ir \, d\langle y, \mu_S(r) x \rangle$$ where $i$ is the imaginary unit.

Let $H$ be a Hilbert space and $\mu_S$ be a spectral measure on $\mathbb{R}$ with values in the operators on $H$. Let $x \in H$ be a vector in the domain of the operator $A = \int_{\mathbb{R}} r \, d\mu_S(r)$ (the maximal spectral integral). For any bounded measurable function $g: \mathbb{R} \to \mathbb{C}$, the squared norm of the difference between the spectral integral of $g$ applied to $x$ and the vector $iAx$ is equal to the integral of the squared difference $|g(r) - ir|^2$ with respect to the diagonal measure $\mu_x$: $$\left\| \left( \int_{\mathbb{R}} g(r) \, d\mu_S(r) \right) x - i Ax \right\|^2 = \int_{\mathbb{R}} |g(r) - i r|^2 \, d\mu_x(r),$$ where $i$ is the imaginary unit and $\mu_x$ is the diagonal measure associated with $x$, defined by $\mu_x(S) = \|\mu_S(S)x\|^2$.

Let $H$ be a Hilbert space and $\mu_S$ be a spectral measure on $\mathbb{R}$ with values in the operators on $H$. Let $e^{itT} = \int_{\mathbb{R}} e^{itr} \, d\mu_S(r)$ be the unitary operator associated with $\mu_S$ for a real parameter $t$. For any $x \in H$, if the scalar difference quotient function $r \mapsto \frac{e^{itr} - 1}{t}$ is bounded by some constant $C$, then the difference quotient of the operator group applied to $x$ satisfies: $$\frac{e^{itT}x - x}{t} = \left( \int_{\mathbb{R}} \frac{e^{itr} - 1}{t} \, d\mu_S(r) \right) x$$

Let $H$ be a Hilbert space and $\mu_S$ be a spectral measure on $\mathbb{R}$ taking values in the operators on $H$. Let $A = \int_{\mathbb{R}} r \, d\mu_S(r)$ be the self-adjoint operator defined by the maximal spectral integral of the identity function. For any vector $x$ in the domain of $A$, the limit of the difference quotient of the unitary group $e^{itA} = \int_{\mathbb{R}} e^{itr} \, d\mu_S(r)$ as $t$ approaches $0$ is given by: $$\lim_{t \to 0, t \neq 0} \frac{e^{itA}x - x}{t} = iAx,$$ where the limit is taken in the norm topology of $H$ and $i$ is the imaginary unit.