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## Research

**Research Interests**

1. Research on extension of Green functions defined for fundamental operators on basic lattice types to operators in the continuous function algebra.

2. Finding Lie groups that permute the solutions of the nonlinear, discrete dynamical systems that come from real-space renormalization and using them to decouple the equations.

3. Simplified theory tools for scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES) of planar molecules on graphene or other surfaces.

4. Finite difference version of exterior calculus applied to lattice models involving coupled fields.

**Selected Reprints**

Scaling and scaling crossover for transport on anisotropic fractal structures

Finding Lie groups that reduce the order of discrete dynamical systems

Group theoretic reduction of Laplacian dynamical problems on fractal lattices

Vector difference calculus for physical lattice models

**List of publications**

Hierarchical analysis of nanostructured materials and systemsInfinitely ramified structures

- "Extension theory for lattice Green functions", W. A. Schwalm and M. K. Schwalm, Phys. Rev. B 37, 9524-9542 (1988).
- "Effects of surface roughness on transport in quasi-one-dimensional and - two-dimensional ballistic channels", M. K. Schwalm and W. A. Schwalm, Phys. Rev. B 45, 6868-6876 (1992).
- "Damping effects on the Kubo-Greenwood conductance of various lattice models in one, two and three dimensions", W. A. Schwalm and M. K. Schwalm, Phys. Rev. B 45, 1770-1775 (1992).
- "Solutions of transport schemes on fractals by means of Green function renormalization--Application to integral quantities", M. Giona, A. Adrover, W. A. Schwalm and M. K. Schwalm, Fractals, 57 473-491(1997).

Kubo-Greenwood conductance and Anderson localization

- "Electronic properties of fractal glass models", W. A. Schwalm and M. K. Schwalm, Phys. Rev. B 39, 12872-12882 (1989).
- "Length scaling of corner-to-corner propagation on fractal lattices," W. A. Schwalm M. K. Schwalm and K. Rada, Phys. Rev. B 44, 382-385 (1991).
- "Perfect conductance on fractal lattices", W. A. Schwalm and M. K. Schwalm, pp. 349-357 in Fractals in the Natural and Applied Sciences, M. M. Novak, editor (Elsevier, Amsterdam, 1994).
- "Length scaling of conductance distribution for random fractal lattices", M. K. Schwalm and W. A. Schwalm, Phys. Rev. B 54, 15086-15093 (1996).

Application of formalism to chemical engineering

- "Exact solution of linear transport equations in fractal media I. Renormalization analysis and general theory" , M. Giona, W. A. Schwalm, M. K. Schwalm and A. Adrover, Chem. Engng. Sci. 51, 4717-4729 (1996).
- "Exact solution of linear transport equations in fractal media II. Diffusion and Convection" M. Giona, W. A. Schwalm, M. K. Schwalm and A. Adrover, Chem. Engng Sci. 51, 4731-4744 (1996).
- "Exact solution of linear transport equations in fractal media III. Adsorption and Chemical Reaction", M. Giona, A. Adrover, W. A. Schwalm and M. K. Schwalm, Chem. Engng Sci. 51, 5065-5076 (1996).
- "Analysis of linear transport phenomena on fractals" M. Giona, W. A. Schwalm, A. Adrover and M. K. Schwalm, The Chemical Engineering Journal 64, 45-61 (1997).
- "First-order kinetics in fractal catalysts: renormalization analysis of the effectiveness factor", M. Giona, W. A. Schwalm, A. Adrover and M. K. Schwalm, Chem. Engng Sci. 51, 2273-2282 (1996).
- "Renormalization analysis of diffusion and adsorption on fractal and disordered lattices in the presence of energetic disorder", M. K. Schwalm, M. Giona, W. A. Schwalm, A. Adrover and M. Giustiniani, Langmuir 13, 1128-1137 (1997).
- "Scaling and scaling crossover for transport on anisotropic fractal lattices", A. Adrover, W. A. Schwalm, M. Giona and D. Bachand, Phys. Rev. E 55, 7304-7312 (1997).
- "Perspectives and application of Green function renormalization to transport phenomena", M. Giona, W. A. Schwalm, M. K. Schwalm and A. Adrover in Fractals and Chaos in Chemical Engineering , M. Giona and G. Biardi eds. (World Scientific, 1997).

Closed formulas for Green functions on fractal lattices

- "Closed formulae for Green functions on fractal lattices," W. A. Schwalm and M. K. Schwalm, Physica A 185, 195-201 (1992).
- "Explicit orbits for renormalization maps for Green functions on fractal lattices", W. A. Schwalm and M. K. Schwalm, Phys. Rev. B 47, 7847-7858 (1993).
- "Explicit Green functions for hierarchical lattices", W. A. Schwalm, C. C. Reese, M. K. Schwalm and C. J. Wagner, Phys. Rev. B 49, 15650-15663 (1994).

Lie groups and reduction of order of renormalization recursions

- "Conic pencils and renormalization dynamics", W. A. Schwalm, C. C. Reese, M. K. Schwalm and C. J. Wagner, Phys. Lett. A 193 238-244 (1994).
- "Lie groups and solution of dynamical problems on fractal lattices", W. A. Schwalm, M. K. Schwalm and M. Giona, in Fractal Frontiers, M. M. Novak and T. G. Dewey eds. (World Scientific, 1997).
- "Group theoretic reduction of Laplacian dynamical problems on regular fractal structures", W. A. Schwalm, M. K. Schwalm and M. Giona, Phys. Rev. E 55 6741-6752 (1997).
- "Group theoretic reduction of linear transport models on regular fractal structures", W. A. Schwalm, M. K. Schwalm and M. Giona in Fractals and Chaos in Chemical Engineering, M. Giona and G. Biardi eds.(World Scientific, 1997).
- "Lie groups and discrete dynamical systems", B. Moritz, W. Schwalm and D. Uherka (accepted, J. Phys. A, Mathematical and General)
- "Finding Lie groups that reduce order of discrete dynamical systems", B. Moritz, W. Schwalm and D. Uherka, J. Phys. A Math. Gen. 31, 7379-7402 (1999).

Vector difference calculus for physical lattices

- "Vector difference calculus for physical lattice models", W. A. Schwalm, B. Moritz, M. Giona and M. K. Schwalm, Phys. Rev. E 59, 1217-1233 (1999).
- "Green function for coupled vector fields", B. Moriz and W. A. Schwalm, J. Phys A:Mth Gen. 34 589-602(2001).

Stiff self-avoiding walks

- "Biased interacting self-avoiding walks on the four-simplex lattice", G. F. Tuthill and W. A. Schwalm, Phys. Rev. B 46, 13722-13734 (1992).