A common route to discreteness in physics is to postulate a Hilbert-space operator and then solveits eigenvalue problem. Here a different, stability-based route is formulated. The starting pointis a variational stability principle: a physically realized stationary state is required not only tosatisfy the stationarity condition δS = 0, but also to be stable with respect to the second variation,δ2S ≥ 0, understood as a positive stability form. Under standard assumptions on the secondvariation–symmetry, closedness, lower semiboundedness, and coercivity after a shift–this form definesa self-adjoint stability operator. If the physical boundary conditions make the relevant embeddingcompact, the stability operator has compact resolvent and therefore a discrete spectrum. In thissense, discreteness is not introduced as an independent quantum postulate; it arises as a spectralconsequence of the second variation together with stability and boundary conditions. The result isstated as a theorem and proved using the representation theorem for closed semibounded forms andcompactness of the Sobolev embedding. The physical meaning is clarified by distinguishing threelevels of discreteness: discrete stability modes, discreteness of action variables, and quantum-typeenergy quantization. Periodic classical systems, including bounded orbital motion, naturally givediscrete stability modes, whereas quantization of energies requires an additional minimal action scaleand a single-valued phase conditi Stability-Induced Discreteness from the Second Variation of Action | Zenodo Skip to main You are using an outdated browser. Please upgrade your browser to improve your experience. There is a newer version of the record available. Published June 2, 2026 | Version v1 Standard Open Stability-Induced Discreteness from the Second Variation of Action Authors/Creators Timur F., Kamalov (Contact person) Description A common route to discreteness in physics is to postulate a Hilbert-space operator and then solve its eigenvalue problem. Here a different, stability-based route is formulated. The starting point is a variational stability principle: a physically realized stationary state is required not only to satisfy the stationarity condition δS = 0, but also to be stable with respect to the second variation, δ2S ≥ 0, understood as a positive stability form. Under standard assumptions on the second variation–symmetry, closedness, lower semiboundedness, and coercivity after a shift–this form defines a self-adjoint stability operator. If the physical boundary conditions make the relevant embedding compact, the stability operator has compact resolvent and therefore a discrete spectrum. In this sense, discreteness is not introduced as an independent quantum postulate; it arises as a spectral consequence of the second variation together with stability and boundary conditions. The result is stated as a theorem and proved using the representation theorem for closed semibounded forms and compactness of the Sobolev embedding. The physical meaning is clarified by distinguishing three levels of discreteness: discrete stability modes, discreteness of action variables, and quantum-type energy quantization. Periodic classical systems, including bounded orbital motion, naturally give discrete stability modes, whereas quantization of energies requires an additional minimal action scale and a single-valued phase condition. Files stability2_induced_discreteness_PRE.pdf Files (380.2 kB) Name Size Download all stability2_induced_discreteness_PRE.pdf md5:88bfd2f15bea20a6e2c95ced94831c81 380.2 kB Preview Download 34 Views 14 Downloads Show more details All versions This version Views Total views 34 9 Downloads Total downloads 14 1 Data volume Total data volume 6.0 MB 760.3 kB More info on how stats are collected.... image:: https://zenodo.org/badge/DOI/10.5281/zenodo.20513283.svg :target: https://doi.org/10.5281/zenodo.20513283 HTML <a href="https://doi.org/10.5281/zenodo.20513283"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.20513283.svg" alt="DOI"></a> Image URL https://zenodo.org/badge/DOI/10.5281/zenodo.20513283.svg Target URL https://doi.org/10.5281/zenodo.20513283 Resource type Standard Publisher Zenodo Rights License Creative Commons Attribution 4.0 International The Creative Commons Attribution license allows re-distribution and re-use of a licensed work on the condition that the creator is appropriately credited.