We prove the existence of the density of states of a local, self-adjoint operator determined by a coercive, almost periodic quadratic form on $H^m\left({ℝ}^{\nu }\right)$. The support of the density coincides with the spectrum of the operator in $L²\left({ℝ}^{\nu }\right)$.

We introduce and study the linear symmetric systems associated with the modified Cherednik operators. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite propagation speed property of it.

Si considera, in uno spazio di Hilbert $H$ l'operatore lineare ${\mathfrak{M}}_0\phi =1/2\text{Tr}\left[D^2\phi \right]+〈x,AD\phi 〉-〈DU\left(x\right),D\phi 〉$, dove $A$ è un operatore negative autoaggiunto e $U$ è un potenziale che soddisfa a opportune condizioni di integrabilità. Si dimostra con un metodo analitico che ${\mathfrak{M}}_0$ è essenzialmente autoaggiunto in uno spazio $L^2\left(H,\nu \right)$ e si caratterizza il dominio della sua chiusura $\mathfrak{M}$ come sottospazio di $W^{2,2}\left(H,\nu \right)$. Si studia inoltre la «spectral gap property» del semigruppo generato da $\mathfrak{M}$.

The characterization of the domain of the Friedrichs extension as a restriction of the maximal domain is well known. It depends on principal solutions. Here we establish a characterization as an extension of the minimal domain. Our proof is different and closer in spirit to the Friedrichs construction. It starts with the assumption that the minimal operator is bounded below and does not directly use oscillation theory.

We are concerned with some unbounded linear operators on the so-called $p$-adic Hilbert space ${𝔼}_{\omega }$. Both the Closedness and the self-adjointness of those unbounded linear operators are investigated. As applications, we shall consider the diagonal operator on ${𝔼}_{\omega }$, and the solvability of the equation $Au=v$ where $A$ is a linear operator on ${𝔼}_{\omega }$.