Abstract The continuous spectrum of a quantum mechanical (QM) system contains important information on the system. However, the normalization of wave functions of the continuous spectrum is often difficult and therefore is often omitted from the books on QM. We discuss this problem and show how we can overcome this problem technically, with some examples.

We present some simple arguments to show that quantum mechanics operators are required to be self-adjoint. We emphasize that the very definition of a self-adjoint operator includes the prescription of a certain domain of the operator. We then use these concepts to revisit the solutions of the time-dependent Schroedinger equation of some well-known simple problems - the infinite square well, the finite square well, and the harmonic oscillator. We show that these elementary illustrations can be enriched by using more general boundary conditions, which are still compatible with self-adjointness. In particular, we show that a puzzling problem associated with the Hydrogen atom in one dimension can be clarified by applying the correct requirements of self-adjointness. We then come to Stone\'s theorem, which is the main topic of this paper, and which is shown to relate the usual definitions of a self-adjoint operator to the possibility of constructing well-defined solutions of the time-dependent Schrödinger equation.

Snake venom (sv) C-type lectins encompass a group of hemorrhagic toxins, which are able to interfere with hemostasis. They share significant similarity in their primary structures with C-type lectins of other animals, and also present a conserved carbohydrate recognition domain (CRD). A very well studied sv C-type lectin is the heterodimeric toxin, convulxin (CVX), from the venoms of South American rattlesnakes, Crotalus durissus terrificus and C. d. cascavella. It consists of two subunits, alfa (CVXalpha , 13.9 kDa) and beta (CVXbeta , 12.6 kDa), joined by inter and intra-chain disulfide bounds, and is arranged in a tetrameric alpha4beta4 conformation. Convulxin is able to activate platelet and induce their aggregation by acting via p62/GPVI collagen receptor. Several cDNA precursors, homolog of CVX subunits, were cloned by PCR homology screening. As determined by computational analysis, one of them, named crotacetin beta subunit, was predicted as a polypeptide with a tridimensional conformation very similar to other subunits of convulxin-like snake toxins. Crotacetin was purified from C. durissus venoms by gel permeation and reverse phase high performance liquid chromatography. The heterodimeric crotacetin is expressed in the venoms of several C. durissus subspecies, but it is prevalent in the venom of C. durissus cascavella. As inferred from homology modeling, crotacetin induces platelet aggregation but noticeably exhibits antimicrobial activity against Gram-positive and Gram-negative bacteria.