# Spectral theory

“Can One Hear the Shape of a Drum?” was asked by Mark Kac in 1966. This was a famous question in the spectral geometry.

The topic of spectral geometry is a broad research area appearing in different mathematical subjects. As such, it allows one to compare spectral information associated with various objects over different domains with selected geometric properties. For example, when the area of the domain is fixed, one often talks of the isoperimetric inequalities in such context.

Geometric spectral inequalities for a collection of most 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. 366pp. linkfree 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
– Dasgupta A., Ruzhansky M., Gohberg lemma, compactness, and essential spectrum of operators on compact Lie groups, J. Anal. Math.,
– Fischer V., Ruzhansky M., Fourier multipliers on graded Lie groupsarxiv
– Akylzhanov R., Ruzhansky M., Lp-Lq multipliers on locally compact groups, J. Funct. Anal., 278 (2020), no. 3, 108324, 49pp. link (open access)arxiv
– Akylzhanov R., Nursultanov E., Ruzhansky M., Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and Lp-Lq Fourier multipliers on compact homogeneous manifoldsJ. Math. Anal. Appl., 479 (2019), 1519-1548. arxivlink
– Akylzhanov R., Majid S., Ruzhansky M., Smooth dense subalgebras and Fourier multipliers on compact quantum groupsComm. Math. Phys., 362 (2018), 761-799. offprint (open access)linkarxiv
– Cardona D., Ruzhansky M., Hormander condition for pseudo-multipliers associated to the harmonic oscillatorarxiv
– Akylzhanov R., Ruzhansky M., Fourier multipliers and group von Neumann algebras, C. R. Acad. Sci. Paris354 (2016), 766-770. offprint (open access)arxivlink
– Ruzhansky M., Wirth J., Lp Fourier multipliers on compact Lie groups, Math. Z.280 (2015), 621-642. offprint (open access)arxivlink
– Ruzhansky M., Wirth J., On multipliers on compact Lie groups, Funct. Anal. Appl., 47 (2013), 72-75.arxivlink

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. linkarxiv
– 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-posednessMath. 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. arxivlink
– Dasgupta A., Ruzhansky M., Eigenfunction expansions of ultradifferentiable functions and ultradistributions, Trans. Amer. Math. Soc.368 (2016), 8481-8498. arxivlink
– Kamotski, I., Ruzhansky, M. Regularity properties, representation of solutions and spectral asymptotics of systems with multiplicities, Comm. Partial Differential Equations32 (2007), 1-35link
– 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 Equations27(2002), 1373-1405. link

Heisenberg oscillator

– Rottensteiner D., Ruzhansky M., The harmonic oscillator on the Heisenberg groupC. R. Acad. Sci. Paris, 358 (2020), 609-615. arxivdoi
– Rottensteiner D., Ruzhansky M., Harmonic and anharmonic oscillators on the Heisenberg grouparxiv