The Main Results

This chapter introduces the objects required for formulating the main results of Lioville-Riemann-Roch type. The results are formulated. In some cases, the Riemann-Roch type equalities cannot be achieved (counterexamples are shown), while inequalities still hold. These inequalities, however, can be applied, the same way the equalities are, for proving the existence of solutions of elliptic equations with prescribed zeros, poles, and growth at infinity.

[1]  M. Kha.  Green's function asymptotics of periodic elliptic operators on abelian coverings of compact manifolds , 2015, 1511.00276.

[2]  Peter Kuchment,et al.  An overview of periodic elliptic operators , 2015, 1510.00971.

[3]  P. Kuchment,et al.  Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral gap interior , 2015, 1508.06703.

[4]  P. Kuchment,et al.  Introduction to Quantum Graphs , 2012 .

[5]  Piotr W. Nowak,et al.  Large Scale Geometry , 2012 .

[6]  P. Kuchment,et al.  Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral edge case , 2011, 1110.0225.

[7]  Jürgen Eichhorn,et al.  Global Analysis on Open Manifolds , 2007 .

[8]  P. Kuchment,et al.  Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds , 2005, math-ph/0503010.

[9]  P. Kuchment,et al.  Integral representations and Liouville theorems for solutions of periodic elliptic equations , 2000, math/0007051.

[10]  M. Gromov,et al.  The Riemann-Roch theorem for elliptic operators , 1993 .

[11]  P. Kuchment.  Floquet Theory for Partial Differential Equations , 1993 .

[12]  Hans L. Cycon,et al.  Schrodinger Operators: With Application to Quantum Mechanics and Global Geometry , 1987 .

[13]  R. Grigorchuk.  Degrees of Growth of Finitely Generated Groups, and the Theory of Invariant Means , 1985 .

[14]  M. Gromov.  Groups of polynomial growth and expanding maps , 1981 .

[15]  L. Saloff-Coste,et al.  Analysis on Riemannian co-compact covers , 2004 .

[16]  F. Klopp,et al.  Endpoints of the spectrum of periodic operators are generically simple , 2000 .

[17]  溝畑 茂,et al.  The theory of partial differential equations , 1973 .

[18]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[19]  Tosio Kato.  Perturbation theory for linear operators , 1966 .