Research » Publications (to download)

- Current Project: Sharp Spectral Asymptotics Reborn
- Talks
- Previous Projects
- Old Papers Digitalized (djvu)
- Some papers written under my supervision

- Microlocal Analysis and Precise Spectral Asymptotics, Springer-Verlag, 1998. (table of contents only; available in your favorite bookstore).
- Sharp Spectral Asymptotics for Operators with Irregular Coefficients.

Internat. Math. Res. Notices 2000, no. 22, 115–1166. - (with M. Bronstein) Sharp Spectral Asymptotics for Operators with Irregular Coefficients. Pushing the Limits

(final version as of Sep. 2002). Comm. Part. Diff. Equats., v. 28, no 1&2, pp. 99–123, (2003). - Sharp Spectral Asymptotics for Operators with Irregular Coefficients. Pushing the Limits II

(final version as of Sep. 2002). Comm. Part. Diff. Equats., v. 28, no 1&2 pp. 125–156, (2003). - Sharp Spectral Asymptotics for Magnetic Schrödinger Operator with Irregular Potential.

(Aug. 2, 2004) Russian Journal of Mathematical Physics, 11:4 (2004), 415–428. - Publications in Bibserver.
- Publications in arXiv.
- Personal page on MathSciNet.

Download them and open via Adobe (Acrobat) Reader.

Do not open via web browser! Some of them are large files!

- Sharp Spectral Asymptotics for Operators with Irregular Coefficients. Pushing the Limits.

My talk at BIRS (March 26, 2003), (Apr 11, 2003) and some other places.

* Streaming video from MSRI (Real Player) - Sharp Spectral Asymptotics for Magnetic Schrödinger Operator.
- Spectral Asymptotics for 2–dimensional Schrödinger Operator with Strong Degenerating Magnetic Field.

My talk at Technion and Weizmann Institute of Science (December 2004). - 25 years after.
- Magnetic Schrödinger Operator: Geometry, Classical and Quantum Dynamics and Spectral Aymptotics.
- Spectral Asymptotics and Dynamics.

* Streaming video from MSRI (QuickTime). - 2D-Magnetic Schröodinger Operator near Boundary.
- 2D- and 3D-Magnetic Schrödinger Operator: Short Loops and Pointwise Spectral Asymptotics.
- 100 years of Weyl' law.
- Large atoms and molecules with magnetic field, including self-generated magnetic field (Results: old, new, in progress and in perspective).
- Some open problems, related to Spectral Theory of PDOs.
- Semiclassical theory with self-generated magnetic field.
- Eigenvalue Asymptotics for Dirichlet-to-Neumann Operator.
- Eigenvalue Asymptotics for Fractional Laplacians.

- e–Articles, e–Books and e–Talks too.
- TeX Freak.
- Beamer All the Way.
- e–Articles, e–Books and e–Talks too (new version; you need Adobe Reader v. 8 or above; extract separate files from Collection (also known as Portfolio).

- Around Scott correction terms.

(Proceedings of the Conference, Saint–Jean–de–Monts, France, June 1994) - Asymptotics of the ground state energies of large Coulomb systems

(with I.M.Sigal). Ann. Math., 138(1993), 243–335. - Semiclassical asymptotics for exchange energy.

Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, Exp. No. XX, 12 pp., École Polytech., Palaiseau, 1994. - Holzhau94.

Semiclassical spectral asymptotics and multiparticle quantum theory (Proceedings of the Conference, Holzhau, Germany, July 1994) - Asymptotics of the ground state energy of heavy molecules in the strong magnetic field.

My talk in Minnesotta May 1995. - Asymptotics of the ground state energy of heavy molecules in the strong magnetic field. I.

(atoms with magnetic field**B=o(N**and molecules with^{3})**B=0**), Russian Journal of Mathematical Physics, 4:1 (1996), 29–74. - Asymptotics of the ground state energy of heavy molecules in the strong magnetic field. II

(molecules with magnetic field**B=o(N**), Russian Journal of Mathematical Physics, 5:3 (1997), 321–354.^{3}) - Heavy molecules in the strong magnetic field.

(Short note: molecules with magnetic field**B=o(N**: asymptotics of the ground state energy, ionization energy estimate, estimates for the excessive negative and positive charges) Russian Journal of Math. Phys.,^{3})**4**, (1996), N 1, 29–74. - Heavy molecules in the strong magnetic field. Estimates for ionization energy and excessive charge.

(molecules with magnetic field**B=o(N**)^{3}) - Heavy molecules in the strong magnetic field

My talk at conference at St.–Jean–de–Monts, 1997 (molecules with magnetic field**B=o(N**).^{3}) - Heavy atoms in the superstrong magnetic field.

My talk at conference at M.Sh.Birman conference, Stockholm, Jan 1998. (Atoms in magnetic field**B>cN**)^{3}### Precise spectral asymptotics for Neumann Laplacian in domains with cusps

- Precise spectral asymptotics for Neumann Laplacian in domains with cusps.
- Eigenvalue asymptotics for Neumann Laplacian in domains with ultra–thin cusps.

(my talk in Ecole Polytechnique, Jan 1999) I deal with Laplacians and similar operators generated by generalized Maxwell system### Miscellaneous

- Accurate Spectral Asymptotics for Periodic Operators.

(Proceedings of the Conference, Saint–Jean–de–Monts, France, June 1999) - Semiclassical spectral asymptotics.

(Proceedings of the Conference, Nantes, France, June 1991)## Old Papers Digitalized (djvu)

Check LizardTech for plug–ins/readers for Windows and MacOS and DjVuLibre for plug–ins/readers for many platforms.

How to digitalize.### Really Old Book

- Precise Spectral Asymptotics for Elliptic Operators Acting in Fiberings over Manifolds With Boundary

Lecture Notes in Mathematics, vol 1100, Springer–Verlag (January 1, 1985).

If you do not have djvu viewers you can still see it via Java applet.### Some of Articles

- Exponential decay of the solution of the wave equation outside an almost star-shaped region, Soviet. Math. Dokl., 10:6 (1989) 1527–1530.
- (with V. Petkov) Necessary Conditions for the Cauchy Problem for Non–Strictly Hyperbolic Equations to be Well–Posed, Russian Math. Surveys, 29:5 (1974), 1–70; Russian
- Well–posedness in Gevrey classes of the Cauchy problem for non–strict hyperbolic operators, Mat. Sbornik, 25:3 (1975), 390–413.
- Sufficient Conditions for Regular and Completely Regular Hyperbolicity, Trudy Moskovskogo Matem. Obshchestva, 33 (1976), 1–65.
- Wave fronts of solutions of certain pseudodifferential equations, Functional Analysis and Its Applications, 10:2 (1976), 141–142.
- Cauchy problem conditions for hyperbolic operators with characteristics of variable multiplicity for Gevrey classes, Siberian Mathematical Journal, 17:6 (1976), 921–931.
- Conditions for correctness in Gevrey classes of the Cauchy problem for weakly hyperbolic equations, Siberian Mathematical Journal, 17:3 (1976), 422–435.
- The Well-Posedness of the Cauchy Problem For Nonstrictly Hyperbolic Operators. III. The Energy Integral, Trudy Moskovskogo Matem. Obshchestva, 34 (1977), 149–168.
- Propagation of Singularities of Solutions of Symmetric Hyperbolic Systems, ICM 1978 (Helsinki), 771–776.
- Wave Front Sets of Solutions of Certain Pseudodifferential Operators, Trudy Moskovskogo Matem. Obshchestva, 39 (1979), 49–86. Also in Russian.
- Wave Front Sets of Solutions of Certain Hyperbolic Pseudodifferential Operators, Trudy Moskovskogo Matem. Obshchestva, 39 (1979), 87–119. Also in Russian.
- Propagation of singularities of solutions of nonclassical boundary–value problems for the wave equation, Functional Analysis and Its Applications, 13:3 (1979), 226–227.
- Wave fronts of solutions of boundary–value problems for symmetric hyperbolic systems, Siberian Mathematical Journal, 20:4 (1979), 516–524.
- Wave fronts of solutions of boundary–value problems for symmetric hyperbolic systems II. Systems with characteristics of constant multiplicity, Siberian Mathematical Journal, 20:5 (1979), 722–734.
- Wave fronts of solutions of symmetric pseudodifferential systems, Siberian Mathematical Journal, 20:3 (1979), 390–405.
- Second term of the spectral asymptotic expansion of the Laplace – Beltrami operator on manifolds with boundary, Functional Analysis and Its Applications, 14:2 (1980), 98–106. Also in Russian.
- Wave fronts for solutions of boundary–value problems for a class of symmetric hyperbolic systems, Siberian Mathematical Journal, 21:4 (1980), 527–534.
- Wave fronts of solutions of boundary–value problems for symmetric hyperbolic systems. III. Systems with characteristics of variable multiplicity, Siberian Mathematical Journal, 21:1 (1980), 54–60.
- The propagation of singularities of solutions of nonclassical boundary value problems for second order hyperbo1jc equations , Trudy Moskovskogo Matem. Obshchestva, 43 (1981), 87–99.
- Exact spectral asymptotics for the Laplace – Beltrami operator in the case of general elliptic boundary conditions, Functional Analysis and Its Applications, 15:1 (1981), 59–60.
- Accurate spectral asymptotics for elliptic operators that act in vector bundles, Functional Analysis and Its Applications, 16:2 (1982), 101–108. (Also in Russian)
- Two papers with O.Zaitseva:
- Correctness of the Cauchy problem for some hyperbolic operators with characteristics of high variable multiplicity, Russian Math. Surveys, 37:3 (1982), 187–188;
- Strict and nonstrict inequalities in conditions for well-posedness of the Cauchy problem, Russian Math. Surveys, 40:2 (1985), 179–180.

- Asymptotics of a spectral problem connected with the Laplace–Beltrami operator on a manifold with boundary, Functional Analysis and Its Applications, 17:1 (1983), 56–57.
- Global and partially global operators. Propagation of singularities and spectral asymptotics. Microlocal analysis (Boulder, Colo., 1983), 119–125, Contemp. Math., 27, Amer. Math. Soc., Providence, RI, 1984.
- Three spectral problems revised. In Hyperbolic equations and related topics (Katata/Kyoto, 1984), pages 85– 88. Academic Press, Boston, MA, 1986.
- Precise eigenvalue asymptotics for transversally elliptic operators In Current topics in partial dirential equations, pages 55–62. Kinokuniya, Tokyo, 1986.
- Asymptotics of the discrete spectrum for certain operators in
**R**, Functional Analysis and Its Applications, 19:1 (1985), 61–62.^{d} - On the number of negative eigenvalues of Schrödinger operators with singular potentials Hyperbolic equations (Padua, 1985), 74–81, Pitman Res. Notes Math. Ser., 158, Longman Sci. Tech., Harlow, 1987.
- (with S.Fedorova) Dilatation and the asymptotics of the eigenvalues of spectral problems with singularities, Functional Analysis and Its Applications, 20:4 (1986), 277–281. (also in Russian)
- Estimates for a Number of Negative Eigenvalues of the Schrödinger Operator with Singular Potentials, ICM 1986, Berkeley, 1084–1093.
- Precise spectral asymptotics for elliptic operators on manifolds with boundary, Siberian Mathematical Journal, 28:1 (1987), 80–86.
- Linear hyperbolic equations (in Russian) Itogi nauli i tehniki. Ser. Sovr. problemy matematiki, Fundamental'nye napravleniya, v. 33, 157–247, 1988, VINITI.
- Spectral asymptotics for the family of commuting operators. Operator calculus and spectral theory (Lambrecht, 1991), 139–148, Oper. Theory Adv. Appl., 57, Birkhauser, Basel, 1992.
- (with C.Fefferman, L.Seco, M.Sigal) The energy asymptotics of large Coulomb systems, Schrödinger operators (Aarhus, 1991), Lecture Notes in Phys., 403, Springer, Berlin, 1992, 79–99.

**Olga Zaitseva**, Six papers on weakly hyperbolic equations, Izvestiya Vysshih Uchebnyh Zavedenij, ser. Matematika, (1980–1987), [in Russian] and [in English].**Mariya Zaretskaya**, Four papers on quadratic quantum Hamiltonians, Izvestiya Vysshih Uchebnyh Zavedenij, ser. Matematika, (1983–1989), [in English]. Note: Translator definitely had no clue about mathematics.