Liouville-Riemann-Roch Theorems on Abelian Coverings

The classical Riemann-Roch theorem has been extended by N. Nadirashvili and then M. Gromov and M. Shubin to computing indices of elliptic operators on compact (as well as non-compact) manifolds, when a divisor mandates a finite number of zeros and allows a finite number of poles of solutions. On the other hand, Liouville type theorems count the number of solutions that are allowed to have a "pole at infinity." Usually these theorems do not provide the exact dimensions of the spaces of such solutions (only finite-dimensionality, possibly with estimates or asymptotics of the dimension. An important case has been discovered by M. Avellaneda and F. H. Lin and advanced further by J. Moser and M. Struwe. It pertains periodic elliptic operators of divergent type, where, surprisingly, exact dimensions can be computed. This study has been extended by P. Li and Wang and brought to its natural limit for the case of periodic elliptic operators on co-compact abelian coverings by P. Kuchment and Pinchover. Comparison of the results and techniques of Nadirashvili and Gromov and Shubin with those of Kuchment and Pinchover shows significant similarities, as well as appearance of the same combinatorial expressions in the answers. Thus a natural idea was considered that possibly the results could be combined somehow in the case of co-compact abelian coverings, if the infinity is "added to the divisor." This work shows that such results indeed can be obtained, although they come out more intricate than a simple-minded expectation would suggest. Namely, the interaction between the finite divisor and the point at infinity is non-trivial.

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

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

[3]  S. Yau,et al.  GEOMETRIC ANALYSIS , 2005 .

[4]  An example of a periodic magnetic Schrödinger operator with degenerate lower edge of the spectrum , 2005 .

[5]  On the Spectra of Carbon Nano-Structures , 2006, math-ph/0612021.

[6]  Michael I. Weinstein,et al.  Honeycomb Lattice Potentials and Dirac Points , 2012, 1202.3839.

[7]  R. Pinsky.  Second Order Elliptic Operators with Periodic Coefficients: Criticality Theory, Perturbations, and Positive Harmonic Functions , 1995 .

[8]  T. Colding,et al.  HARMONIC FUNCTIONS ON MANIFOLDS , 1997 .

[9]  B. Riemann,et al.  Theorie der Abel'schen Functionen. , 1857 .

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

[11]  W. Lueck.  Survey on geometric group theory , 2008, 0806.3771.

[12]  Rachel J. Steiner,et al.  The spectral theory of periodic differential equations , 1973 .

[13]  F. Ledrappier,et al.  Analysis on Riemannian co-compact covers , 2006 .

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

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

[16]  Kaoru Ono,et al.  Periodic Schrödinger Operators on a Manifold , 1989 .

[17]  Patrick J. Morandi.  Field and Galois theory , 1996 .

[18]  Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral edge case , 2012 .

[19]  P. Diaconis,et al.  Convolution powers of complex functions on Z , 2012, 1212.4700.

[20]  M. Kha,et al.  Polynomial-like elements in vector spaces with group actions , 2018, Contemporary Mathematics.

[21]  G. Roch,et al.  Ueber die Anzahl der willkürlichen Constanten in algebraischen Functionen. , 1865 .

[22]  V. Volpert.  Index of Elliptic Operators , 2011 .

[23]  I. M. Glazman.  Direct methods of qualitative spectral analysis of singular differential operators , 1965 .

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

[25]  C. Wilcox.  Theory of Bloch waves , 1978 .

[26]  Harmonic functions with a given set of singularities , 1988 .

[27]  J. Lions,et al.  Un théorème de Liouville pour des équations elliptiques à coefficients périodiques , 1989 .

[28]  Threshold Effects near the Lower Edge of the Spectrum for Periodic Differential Operators of Mathematical Physics , 2001 .

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

[30]  I. Gel'fand.  on Elliptic Equations , 1960 .

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

[32]  Kristian Kirsch,et al.  Methods Of Modern Mathematical Physics , 2016 .

[33]  Asymmetric fermi surfaces for magnetic schrödinger operators , 2000 .

[34]  J. Moser,et al.  On a Liouville-type theorem for linear and nonlinear elliptic differential equations on a torus , 1992 .

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

[36]  P. Kuchment,et al.  GREEN ’ S FUNCTION ASYMPTOTICS NEAR THE INTERNAL EDGES OF SPECTRA OF PERIODIC ELLIPTIC OPERATORS , 2015 .

[37]  G. V. Chester,et al.  Solid State Physics , 2000 .

[38]  Peter Li,et al.  Polynomial growth solutions of uniformly elliptic operators of non-divergence form , 2001 .

[39]  S. Yau.  Nonlinear Analysis In Geometry , 1986 .

[40]  P. Kuchment,et al.  BANACH BUNDLES AND LINEAR OPERATORS , 1975 .

[41]  Gregory Berkolaiko,et al.  Symmetry and Dirac points in graphene spectrum , 2014, Journal of Spectral Theory.

[42]  Barry Simon,et al.  Comparison theorems for the gap of Schrödinger òperators , 1987 .

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

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

[45]  A. Dujella.  Elliptic equations , 2009 .

[46]  Stability in the Gromov-Shubin index theorem , 1997 .

[47]  L2 Riemann-Roch Theorem for Elliptic Operators , 1995 .

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

[49]  On elliptic operators in , 1980 .