Spectral geometry studies how the spectrum of differential operators reflects the geometry of the underlying space. A central theme is how spectral information relates to geometric quantities such as the area of a domain, leading to isoperimetric inequalities and related estimates.

Geometric spectral inequalities for important differential and integral operators

• M. Ruzhansky, M. Sadybekov, D. Suragan, Spectral geometry of partial differential operators, Monographs and Research Notes in Mathematics, Chapman and Hall/CRC Press, 2020, 366 pp. Publisher; free download.

Fourier and spectral multipliers in different settings

• Cardona D., Ruzhansky M., Littlewood-Paley theorem, Nikolskii inequality, Besov spaces, Fourier and spectral multipliers on graded Lie groups. Potential Analysis, 60 (2024), 965–1005.
• Cardona D., Chatzakou M., Ruzhansky M., Toft J., Schatten-von Neumann properties for Hörmander classes on compact Lie groups. Forum Math., 38 (2026), 1335–1361.
• Akylzhanov R., Ruzhansky M., L^p–L^q multipliers on locally compact groups. J. Funct. Anal., 278 (2020), no. 3, 108324.
• Akylzhanov R., Nursultanov E., Ruzhansky M., Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and L^p–L^qFourier multipliers on compact homogeneous manifolds. J. Math. Anal. Appl., 479 (2019), 1519–1548.
• Akylzhanov R., Majid S., Ruzhansky M., Smooth dense subalgebras and Fourier multipliers on compact quantum groups. Comm. Math. Phys., 362 (2018), 761–799.

Wave equation for operators with discrete spectrum

• Ruzhansky M., Tokmagambetov N., On nonlinear damped wave equations for positive operators. I. Discrete spectrum. Differential Integral Equations, 32 (2019), 455-478.
• Ruzhansky M., Tokmagambetov N., Wave equation for operators with discrete spectrum and irregular propagation speed, Arch. Ration. Mech. Anal., 226 (2017), 1161-1207. offprint (open access), arxiv, link

Spectral identities, spectral shift function, spectral asymptotics for operators

• Ben-Artzi M., Ruzhansky M., Sugimoto M., Spectral identities and smoothing estimates for evolution operators. Adv. Diff. Equations, 25 (2020), 627-650. link, arxiv 
• Garetto C., Jäh C., Ruzhansky M., Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, II: microlocal analysis. J. Differential Equations, 269 (2020), 7881-7905. arxiv, link (open access)
• Garetto C., Jäh C., Ruzhansky M., Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness, Math. Ann., 372 (2018), 1597–1629. offprint (open access), arxiv, link
• Delgado J., Ruzhansky M., Wang B., Grothendieck-Lidskii trace formula for mixed-norm and variable Lebesgue spaces, J. Spectr. Theory, 6 (2016), 781-791. arxiv, link
• Dasgupta A., Ruzhansky M., Eigenfunction expansions of ultradifferentiable functions and ultradistributions, Trans. Amer. Math. Soc., 368 (2016), 8481-8498. arxiv, link
• Kamotski, I., Ruzhansky, M. Regularity properties, representation of solutions and spectral asymptotics of systems with multiplicities, Comm. Partial Differential Equations, 32 (2007), 1-35. link
• Pushnitski, A., Ruzhansky, M. Spectral shift function of the Schrodinger operator in the large coupling constant limit, Funct. Anal. Appl., 36 (2002), 93-95. offprint, link
• Pushnitski, A., Ruzhansky, M. Spectral shift function of the Schrodinger operator in the large coupling constant limit, II. Positive perturbations, Comm. Partial Differential Equations, 27(2002), 1373-1405. link

Heisenberg oscillator

• Rottensteiner D., Ruzhansky M., The harmonic oscillator on the Heisenberg group. C. R. Acad. Sci. Paris, 358 (2020), 609-615. arxiv, doi
• Rottensteiner D., Ruzhansky M., Harmonic and anharmonic oscillators on the Heisenberg group. arxiv