Abstract:
As shown in Ref. \cite{Rod09}, the Fermi part $M_{F}^{0\nu}$ of the total $0\nu\beta\beta$-decay nuclear matrix element $M^{0\nu}$ can be related to the single Fermi transition matrix element between the isobaric analog state (IAS) of the ground state of the initial nucleus and the ground state of the final nucleus. The latter matrix element could be measured in charge-exchange reactions. Here we discuss a possibility of such a measurement for $^{48}$Ca and estimate the cross-section of the reaction $^{48}$Ti(n,p)$^{48}$Sc(IAS).

Abstract:
The present status of calculations of the nuclear matrix elements for neutrinoless double beta decay is reviewed. A proposal which allows in principle to measure the neutrinoless double beta decay Fermi matrix element is briefly described.

Abstract:
Some aspects, both experimental and theoretical, of extracting the neutron skin $\Delta R$ from properties of isovector giant resonances are discussed. Existing proposals are critically reviewed. The method relying on the energy difference between the GTR and IAS is shown to lack sensitivity to $\Delta R$. A simple explanation of the linear relation between the symmetry energy and the neutron skin is also given.

Abstract:
After application of an analytical transformation, a new exact representation for the nuclear isospin-symmetry breaking correction $\delta_C$ to superallowed beta decay is obtained. The correction is shown to be essentially the reciprocal of the square of an energy parameter $\Omega_M$ which characterizes the charge-exchange monopole strength distribution. The proportionality coefficient in this relation is determined by basic properties of the ground state of the even-even parent nucleus, and should be reliably calculable in any realistic nuclear model. Therefore, the single parameter $\Omega_M$ contains all the information about the properties of excited $0^+$ states needed to describe $\delta_C$. This parameter can possibly be determined experimentally by charge-exchange reactions. Basic quantities of interest are calculated within the isospin-consistent continuum random phase approximation, and the values of $\delta_C$ are compared with the corresponding results from other approaches.

Abstract:
A method to calculate the nuclear double beta decay ($2\nu\beta\beta$- and $0\nu\beta\beta$-) amplitudes within the continuum random phase approximation (cQRPA) is formulated. Calculations of the $\beta\beta$ transition amplitudes within the cQRPA are performed for ^{76}Ge, ^{100}Mo and ^{130}Te. A rather simple nuclear Hamiltonian consisting of phenomenological mean field and zero-range residual particle-hole and particle-particle interaction is used. The calculated M^{2\nu} are almost not affected when the single-particle continuum is taken into account. At the same time, a regular suppression of the $0\nu\beta\beta$-amplitude is found that can be associated with additional ground state correlations due to collective states in the continuum. It is expected that future inclusion of the nucleon pairing in the single-particle continuum will somewhat compensate the suppression.

Abstract:
By making use of the isospin conservation by strong interaction, the Fermi $0\nu\beta\beta$ nuclear matrix element $M_{F}^{0\nu}$ is transformed to acquire the form of an energy-weighted double Fermi transition matrix element. This useful representation allows reconstruction of the total $M_{F}^{0\nu}$ provided a small isospin-breaking Fermi matrix element between the isobaric analog state in the intermediate nucleus and the ground state of the daughter nucleus could be measured, e.g. by charge-exchange reactions. Such a measurement could set a scale for the $0\nu\beta\beta$ nuclear matrix elements and help much to discriminate between different nuclear structure models in which calculated $M_{F}^{0\nu}$ may differ by as much as the factor of 5.

Abstract:
In the present work the sensitivity of the QRPA calculation results to a realistic residual interaction is analyzed in the framework of the approach of Refs. \cite{Rum98,Rodin05}. Both Gamow-Teller (GT) and Fermi (F) \bb-decay amplitudes $M^{2\nu}$, along with the corresponding energy-weighted sum rules $S$, are calculated. General expressions relating $S$ to a realistic residual particle-particle interaction are derived, which show a pronounced sensitivity of $S$ to the singlet-channel interaction in the case of F transitions, and to the triplet-channel interaction in the case of GT transitions. Decompositions of $M^{2\nu}$, as well as the monopole transition contributions to $M^{0\nu}$, are obtained by the method of Refs. \cite{Rum98,Rodin05}. It is shown that in most of the cases almost the whole dependence of $M^{2\nu}$ and $M^{0\nu}$ on the particle-particle renormalization parameter $g_{pp}$ is accounted for by the $g_{pp}$-dependence of the corresponding sum rules $S$. Thus, the $g_{pp}$-sensitivity of calculated $M^{2\nu}$ and $M^{0\nu}$ is unavoidable since it is dictated by the generic structure of the $\beta\beta$ amplitudes. Finally, a better isospin-consistent way of a renormalization of the realistic residual particle-particle interaction to use in QRPA calculations is suggested.

Abstract:
A continuum-QRPA approach to calculation of the $2\nu\beta\beta$- and $0\nu\beta\beta$-amplitudes has been formulated. For $^{130}$Te a regular suppression (about 20%) of the high-multipole contributions to the $0\nu\beta\beta$-amplitude has been found which can be associated with additional ground state correlations appearing from the transitions to collective states in the continuum. At the same time the total calculated $0\nu\beta\beta$-amplitude for $^{130}$Te gets suppressed by about 20% as compared to the result of the usual, discretized, QRPA.

Abstract:
As shown in Ref.\cite{Rod09}, the Fermi nuclear matrix element $M^{0\nu}_F$ of neutrinoless double beta ($0\nu\beta\beta$) decay can be reconstructed if one is able to measure the isospin-forbidden single Fermi transition matrix element from the ground state of the final nucleus to the isobaric analog state (IAS) of the initial nucleus, for instance by means of charge-exchange reactions of the $(n,p)$-type. Here, simple estimates for $^{82}$Se are made which show that indeed the tiny cross section $\sigma_{np}(0_f^+\to IAS)$ is dominated by the admixture of the double IAS in the ground state of the final nucleus provided that the isospin mixing is weak and can be treated perturbatively. A measurement of such a cross section would definitely be a very difficult task, but it can advance a lot our knowledge of the $0\nu\beta\beta$ nuclear matrix element.

Abstract:
The renormalized quasiparticle-RPA is reformulated for even-even nuclei using restrictions imposed by the commutativity of the phonon creation operator with the total particle number operator. This new version, Fully-Renormalized QRPA (FR-QRPA), is free from the spurious low-energy solutions. Analytical proof is given that the Ikeda sum rule is fullfiled within the FR-QRPA.