This is the conclusion of the series of 5 volumes. We addess to the reviews of the preceding 4 volumes for details (see [Zbl 1441.35002], [Zbl 1441.35003], [Zbl 1443.35003], [Zbl 1445.35007]), noting in particular that the first two volumes are devoted to the semiclassical methods applied here to quantum theory. In fact, the first part of the present Volume 5 concerns multiparticle quantum Hamiltonians and Thomas-Fermi theory. Particular attention is devoted to asymptotics for the ground state energy, that is the lowest eigenvalue of the operator. Namely, in Chapter 25, first of this volume, the author studies in detail the case when the electrons interact between themselves without magnetic field. The precise analysis involves the Scott and the Dirac-Schwinger correction terms, negatively and positively charged systems and internuclear energy. Chapter 26 is devoted to the case of external magnetic field, the corresonding one-particle Hamiltonian being the Schrödinger-Pauli operator. Chapter 27 concerns the case of a self-generated magnetic field. The final Chapter 28 is devoted to the case when the two previous types of magnetic fields combine.
The second part of the volume is somewhat non-standard for a monography. It consists of a list of papers produced by Ivrii very recently. They improve some contents of the initial volumes of the series and reflect new interests of research of the author.
Let us add in conclusion some comments concerning the whole book. The work of Ivrii is impressive. The 5 volumes give a total of about 4000 pages, covering a large part of the subject of the spectral asymptotics for partial differential equations. It took weeks for the reviewer to give a glance to the contents, and it would take years to the reader for a serious study of the text, including the solutions of many interesting problems suggested by the author. Setting the role of the book in the modern literature, one notes the connection between classical results in spectral theory, due mainly to the Russian school, and recent methods of microlocal and semiclassical analysis. This is evident from the Bibliography, where the main part of the contributions in classical theory belong to M. S. Birman, A. Laptev, M. Z. Solomyak, M. A. Shubin, G. Raikov, G. V. Rozenblioum, Yu. Safarov, A. V. Sobolev and D. G. Vassiliev. On the other hand, the microlocal perspective is not neglected, see the contributions of Y. Colin de Verdiére, M. Dimassi, G. Grubb, B. Helffer, R. Melrose, A. Parmeggiani, J. Sjostrand, D. Robert, H. Tamura and M. E. Taylor. Beside the description of the interplay between different schools, the relevance of the book resides in the detailed presentation of the excellent results of Ivrii on sharp spectral asymptotics, and in the wealth of the material.