Misha Shubin. 1944 -- 2020
M. Braverman, B. M. Buchshtaber, M. Gromov, V. Ivrii, Yu. A. Kordyukov, P. Kuchment, V. Maz'ya, S. P. Novikov, T. Sunada, L. Friedlander, A. G. Khovanskii
MMisha Shubin1944 – 2020Mikhail Aleksandrovich Shubin, an outstanding mathematician, passedaway after a long illness on May 13, 2020. Mikhail Aleksandrovichwas born on December 19, 1944 in Kujbyshev (now Samara). Hewas brought up by his mother and grandmother. His mother, MariaArkadievna, was an engineer at the State Bearing Factory, where shewas hired in 1941 after having graduated from the Department ofMechanics and Mathematics of the Moscow State University (MSU).At that time, the factory was evacuated from Moscow to Kujbyshev.Maria Arkadievna had been working at the factory for many years asthe Head of the Physics of Metals Laboratory. Later, she defended aPhD dissertation and moved to the Kujbyshev Polytechnical Institute,where she worked as an Associate Professor.In his school years, Shubin was mostly interested in music. He hadan absolute pitch, finished a music school, and was seriously thinking ofentering a conservatory. However, while in high school, he developed aninterest to mathematics, was successful in olympiads, and eventuallydecided to apply to the Department of Mechanics and Mathematics(mech-mat).Mikhail Aleksandrovich was admitted to mech-mat of MSU in 1961.When the time came to choose an adviser, he became a pupil of MarkoIosifovich Vishik. After graduation, he was enrolled to the GraduateProgram, and in 1969 defended his PhD Thesis. In the dissertation a r X i v : . [ m a t h . HO ] A ug he derived formulas for computing the index of matrix-valued Wiener-Hopf operators. In particular, for the study of families of such opera-tors, he had to generalize a theorem of Birkhoff stating that a continu-ous matrix-valued function M ( z ) defined on the unit circle | z | = 1 canbe factored as M ( z ) = A + ( z ) D ( z ) A − ( z ) , where A + ( z ) and A − ( z ) are continuous and have analytic continuationsto the interior of the unit circle and its exterior (infinity included),respectively; D ( z ) is a diagonal matrix with the entries z n j on thediagonal, with integer n j . Shubin considered the problem about whathappens when the matrix M depends continuously on an additionalparameter t . The Birkhoff factorization can not be made continuous in t . Mikhail Aleksandrovich showed that if one relaxes the condition that D ( z ) is diagonal, allowing it to be just triangular, then the factorizationcan be done continuously with respect to t .Another paper [13] from that period deserves to be mentioned. Leta family of closed subspaces of a Banach space depend analyticallyon a multi-dimensional complex parameter. This means that locallywith respect to the parameter there exists a holomorphic projector-valued function, with the subspaces from the family being the rangesof the corresponding projections. One wonders whether there existssuch a globally holomorphic projector-valued function. Shubin provedthat if the domain of the parameter is holomorphically convex (or amore general Stein complex space), the answer is positive. That re-quired application of a recent result of L. Bungart, which generalizedto an infinite-dimensional situation a difficult and not old at that timetheorem by H. Grauert claiming that holomorphic and topological clas-sifications of vector bundles over Stein spaces coincide. This result ofShubin has become classical and extensively used in the operator the-ory. The paper was one of early manifestations of the broad range ofMikhail Aleksandrovich’s knowledge.The notion of a pseudodifferential operator was formalized in themid-sixties, and the rapid development of the micro-local analysis en-sued. The micro-local analysis, one way or another became central topractically all works by Shubin. In collaboration with V. N. Tulovsky[9], he studied the spectral asymptotics of an operator P of order m > R n . The spectrum of this operator is discrete,if the symbol satisfies the following conditions:(1) | D α p ( y ) | ≤ C α (1 + | y | ) m − ρ | α | , where 0 < ρ ≤ | D α p ( y ) | ≤ C α p ( y ) | y | − ρ | α | , and(3) p ( y ) ≥ a | y | m for large | y | , where the numbers a and m are positive. Under someadditional assumptions about the symbol, they proved that when λ goes to infinity, the number of the eigenvalues that do not exceed λ asymptotically equals (2 π ) − n times the measure of the set { y : p ( y ) <λ } . They also obtained a remainder estimate. The method of theproof (which has became popular since) was remarkable: they appliedtechniques of micro-local analysis to construct approximate spectralprojections, and then used variational methods. The class of operatorswith symbols satisfying (1) is frequently called the Shubin class.In the second half of seventies, Mikhail Aleksandrovich worked mostlyon spectral theory of operators with almost periodic and random co-efficients. He introduced a class of pseudodifferential operators withalmost periodic symbols and studied spectral properties of selfadjointoperators from this class. He also proved existence of the density ofstates and of the Fermi energy for such operators. Let A be an ellipticselfadjoint almost periodic differential operator in R n and let Ω k be afamily of ”good” (e.g., with smooth boundary) increasing domains thatexhaust the space and such that the ratio of their surface areas to thevolumes goes to zero as k increases (so-called F¨olner condition). Con-sider the operator A in Ω k with selfadjoint elliptic boundary conditions(e.g., Dirichlet). Let E j ( k ) be its eigenvalues taken in non-decreasingorder and counted with their multiplicities. Let N k ( E ) be the numberof these eigenvalues that are less than E . The density of states is thelimit of N k ( E ) / | Ω k | when k → ∞ , and the Fermi energy E F ( ρ ) is thelimit 1 p ( k ) (cid:0) E ( k )1 + . . . + E ( k ) p ( k ) (cid:1) , where k → ∞ and p ( k ) / | Ω k | → ρ .These works constituted his Doctor of Science (the degree in Russiaroughly equivalent to Habilitation) Dissertation, which he defended in1981 in Leningrad. There Shubin also developed functional calculus ofpseudodifferential operators with almost periodic symbols, proved thatthe spectrum in different function spaces is the same, that the spectral ζ function admits analytic continuation, and derived a formula for theindex [14–17, 19, 20]. In particular, he used the II ∞ -factor introducedby Coburn, Moyer and Singer. The methods used for studying operators with almost periodic co-efficients are reminiscent of those that are known for operators withrandom coefficients. Following this connection, in collaboration withB. V. Fedosov [10], Shubin derived a formula for the index of ellipticoperators in R n with homogeneous random fields as coefficients. Be-cause these operators are not Fredholm in the standard sense, theirindex is defined in terms of traces in von Neumann algebras. Togetherwith S. M. Kozlov [5, 6], they studied the problem of equality of thespectra in different functional spaces.Shubin has made also a significant contribution to the theory ofoperators with periodic coefficients. In the joint work with D. Schenk[11,12] they derived a complete asymptotic expansion (when the energygoes to infinity) of the density of states for the Hill operator witha smooth potential. One of the difficulties here came from the factthat for a generic potential all gaps in the spectrum are open, andthus the expansion can not be differentiable. Only recently, the multi-dimensional analog of this result has been obtained by L. Parnovskiand R. Shterenberg.In 1982 E. Witten introduced a deformation of the de Rham complexassociated with a Morse function on a closed manifold, and used it fora new proof of the Morse inequalities. S. P. Novikov and M. A. Shu-bin [7] noticed that the Witten’s deformation can be applied to thefollowing situation. Let M be a closed manifold with an infinite fun-damental group, and let (cid:102) M be its universal covering space. Introducea Riemannian metric on M and lift it to (cid:102) M . Consider the Laplacian∆ p acting on p -forms on (cid:102) M , and let K p ( x, y ) be the Schwartz kernel ofthe orthogonal projection onto the null-space of ∆ p . The fundamentalgroup π ( M ) acts on (cid:102) M , and K p ( x, y ) is invariant with respect to thisaction. One can define L -Betti numbers as follows: β p ( M ) := (cid:90) F tr K p ( x, x ) dx, where F is a fundamental domain of the action of π ( M ) on (cid:102) M . Novikovand Shubin proved that if f ( x ) is a Morse function on M , then theMorse inequalities remain valid if one replaces the usual Betti numbersby their L analogs. They also introduced a new system of invariantsof a manifold that was named later the Novikov–Shubin invariants.Let N p ( λ, x, y ) be the Schwartz kernel of the spectral projection for theoperator ∆ p , and let(4) N p ( λ ) := (cid:90) F tr N p ( λ, x, x ) dx. The invariant α p is the smallest number, for which N p ( λ ) − β p ( M ) = O ( λ α p )as λ →
0. Later, Shubin and M. Gromov showed [27] that α p are ho-motopic invariants of M . These papers with Novikov and with Gromovtreat also a more general case of a representation of the fundamentalgroup π ( M ) in a II -factor.The classical Riemann–Roch theorem can be interpreted as a for-mula for the change of the index of the Cauchy–Riemann operator thatoccurs when one enforces a divisor of zeros and singularities. Some gen-eralizations of this theorem to elliptic operators were obtained in theworks by V. G. Maz’ya and B. A. Plamenevski and N. S. Nadirashvili.M. A. Shubin, in collaboration with M. Gromov, obtained far reachinggeneralization of these theorems for elliptic operators acting on sectionsof vector bundles over closed manifolds, as well as for the “divisors”that are not necessarily discrete. We will formulate one of their results.Let A be an elliptic differential operator acting on sections of a vectorbundle E over a closed manifold M ; E ∗ is the dual bundle to E and A ∗ is the transposed operator to A . A rigged divisor is a quadruple( D + , L + ; D − , L − ) where D + , D − are disjoint nowhere dense closed setsin M , and L + , L − are finite-dimensional spaces of distributions withthe values in E and E ∗ and with supports lying in D + and D − respec-tively. By L ( µ, A ) we denote the space of smooth sections of E over M \ D + that can be extended to distributions u over M with values in E such that u is orthogonal to L − , and Au ∈ L + . Then(5) dim L ( µ, A ) = ind A + deg A ( µ ) + dim L ( µ − , A ∗ ) . Here µ − := ( D − , L − ; D + , L + ) and deg A ( µ ) is some “degree” of therigged divisor (usually explicitly computable), which we will not definehere. Note that if one writes the above equality asdim L ( µ, A ) − dim L ( µ − , A ∗ ) = ind A + deg A µ, A, it can be interpreted as the perturbation of the index occurring whenone enforces a divisor of “zeros and singularities.”Later, Shubin obtained an L -version of this result for non-compactregular coverings of compact manifolds.In the second half of the 90s and in the beginning of 2000s, Shu-bin extensively studied, both on his own and in collaboration withV. G. Maz’ya and V. A. Kondratiev, criteria of discreteness of the spec-trum and of essential self-adjointness for Schr¨odinger operators withelectric and magnetic potentials [33–37]. These papers are related tothe following result of A. M. Molchanov that was obtained in 1953. Let H = − ∆ + V ( x ) be a Schr¨odinger operator in R n with a locallyintegrable potential V ( x ). A. M. Molchanov showed in 1953 that thefollowing condition is equivalent to the discreteness of the spectrum ofthe operator H : for every d > (cid:90) Q d \ F V ( x ) dx → ∞ when Q d → ∞ . Here Q d is a cube of side d , the sides of which areparallel to the co-ordinate axes, and the infimum is taken over compactsets F such that their Wiener capacity cap F does not exceed γ cap Q d (such sets are called γ -negligible), and γ is small enough. Giving ananswer to I. M. Gel’fand’s question, Maz’ya and Shubin showed thatevery γ such that 0 < γ < γ dependon d in such a way that lim sup d → γ ( d ) d − = ∞ . For operators with a magnetic potential, the discreteness of the spec-trum criteria were obtained in terms of an “effective” scalar potentialintroduced for that purpose.In the paper “Can one see the fundamental frequency of a drum?”[38] Maz’ya and Shubin obtained a beautiful and far reaching gener-alization of the following result by W. K. Heiman: let Ω be a simplyconnected domain in R and let r (Ω) be the maximal radius of a circleinscribed in Ω (“inner radius”). Then the smallest eigenvalue of theDirichlet Laplacian in Ω satisfies the estimate cr − (Ω) ≤ λ ( ω ) ≤ Cr − (Ω)with absolute constants c and C . This result does not hold for non-simply connected domains (drilling a large number of small holes in adomain gives a counter-example), and it also does not hold in higherdimensions. Maz’ya and Shubin showed that if one changes the defini-tion of the inner radius appropriately, then one can extend the resultto arbitrary domains in arbitrary dimension n . Namely, let 0 < γ < r γ (Ω) be the maximal radius of a closed ball B r such that B r \ Ωis γ -negligible in B r . Then c ( n, γ ) r − γ (Ω) ≤ λ ( ω ) ≤ C ( n, γ ) r − γ (Ω) . Mikhail Aleksandrovich had also worked in a variety of other areasof mathematics. One should mention in particular his undeservedlylittle known paper on pseudo-difference operators and on the estimatesof discrete Green’s functions [22] (see also [40, 44]). Nowadays, discretespectral problems attract a lot of attention, and Shubin’s technique should be extremely useful. It is impossible to adequately describe inthis article his numerous papers (both single authored and joint with hisstudents and collaborators) devoted to the spectral theory of operatorson non-compact manifolds and on Lie Groups (in particular, see [2, 21,39,41]). He had also worked on completely integrable equations [31,32],applications of non-standard analysis [4], and other problems. Theprobably incomplete list of his publications in MathSciNet contains135 items.Shubin has written several remarkable books. Published in 1978, hisbook ”Pseudodifferential operators and spectral theory” [18] instantlybecame one of the main textbooks on microlocal analysis. Now, aftermore than forty years, it is still very popular. His joint monographwith F. A. Berezin ”The Schr¨odinger Equation” [1] has been very in-fluential. In that book, which was finished after Felix AlaksandrovichBerezin passed away, many topics, e.g. the path integrals, are treatedin a rather unique way. In 2001 Mikhail Aleksandrovich wrote the text-book “Lectures on the equations of mathematical physics” [23], whichwas based upon the lectures that he gave at the Moscow State Uni-versity. Somewhat extended English translation of this book, editedby M. Braverman, R.McOwen, and P. Topalov, was recently publishedby the AMS [43]. In addition to numerous expository articles that hewrote for Russian Math. Uspehi, one should also mention extensive sur-veys that he wrote in collaboration with other authors for the VINITIpublications (for example, [3, 8]). He also edited Russian translationsof the fundamental books by F. Treves and L. H¨ormander.Mikhail Aleksandrovich started participating in the famous seminarof I. M. Gel’fand in October of 1964, when he was a junior undergrad-uate student. He had been taking notes, practically with no interrup-tions, for 25 years, and preserved all of them. Now, with the financialsupport from the Clay Institute, these notes are available on the in-ternet as the “Gelfand Seminar notes” file [25]. Gelfand’s seminar wasthe center of mathematical life in the USSR, and Shubin’s notes havebecome a unique testament of that years.Mikhail Aleksandrovich was a remarkable lecturer and teacher. Hislectures were extremely well prepared, with all small wrinkles ironedout. In a rather short period if time, he would lead students to non-trivial problems. He had advised on about 20 PhD dissertations. Inaddition to his PhD students, he strongly influenced many other youngmathematicians. This was not limited to undergraduate and graduatestudents. He had worked a lot with high school students in summermath camps; as a result he published some of his lecture notes there and a small book [24] in the “Mathematics Education” series. Start-ing 1969, Shubin worked at the Division of Differential Equations ofthe Department of Mechanics and Mathematics of the Moscow StateUniversity. In 1992 he became a Professor, and then DistinguishedProfessor of the Department of Mathematics of the Northeastern Uni-versity in Boston, USA.Mikhail Aleksandrovich was a remarkable, broadly educated, cheer-ful, and friendly person. In the seventies and eighties he was helpingseveral Soviet mathematicians who had difficulties due to political rea-sons. He also fought against unfair admission practices that were takingplace at the Department of Mathematics of the MSU.The memory of Mikhail Aleksandrovich Shubin, a mathematician,teacher, and a human being, will remain forever in the hearts of hisnumerous colleagues, students, and friends.
M. Braverman, B. M. Buchshtaber, M. Gromov, V. Ivrii,Yu. A. Kordyukov, P. Kuchment, V. Maz’ya, S. P. Novikov,T. Sunada, L. Friedlander, A. G. Khovanskii
References [1] Berezin, F. A. ; Shubin, M. A. The Schrdinger equation. Translated from the1983 Russian edition by Yu. Rajabov, D. A. Letes and N. A. Sakharova andrevised by Shubin. With contributions by G. L. Litvinov and Letes. Mathe-matics and its Applications (Soviet Series), 66. Kluwer Academic PublishersGroup, Dordrecht, 1991. xviii+555 pp. ISBN: 0-7923-1218-X[2] M. Braverman, O. Milatovich, M. Shubin, Essential selfadjointness ofSchr¨odinger-type operators on manifolds. (Russian) ; translated from UspekhiMat. Nauk 57 (2002), no. 4(346), 3–58 Russian Math. Surveys 57 (2002), no.4, 641–692[3] Egorov, Yu. V. ; Shubin, M. A. Linear partial differential equations. Elementsof the modern theory [MR1175406 (93e:35001)]. Partial differential equations,II, 1–120, Encyclopaedia Math. Sci., 31, Springer, Berlin, 1994.[4] Zvonkin, A. K. ; Shubin, M. A. Nonstandard analysis and singular pertur-bations of ordinary differential equations. (Russian) Uspekhi Mat. Nauk 39(1984), no. 2(236), 77–127.[5] Kozlov, S. M. ; Shubin, M. A. A theorem on the coincidence of spectra forrandom elliptic operators. (Russian) Funktsional. Anal. i Prilozhen. 16 (1982),no. 4, 74–75.[6] Kozlov, S. M. ; Shubin, M. A. Coincidence of spectra of random elliptic oper-ators. (Russian) Mat. Sb. (N.S.) 123(165) (1984), no. 4, 460–476.[7] Novikov, S. P. ; Shubin, M. A. Morse inequalities and von Neumann II -factors.(Russian) Dokl. Akad. Nauk SSSR 289 (1986), no. 2, 289–292.[8] Rozenblyum, G. V. ; Solomyak, M. Z. ; Shubin, M. A. Spectral theory of dif-ferential operators. (Russian) Current problems in mathematics. Fundamentaldirections, Vol. 64 (Russian), 5–248, Itogi Nauki i Tekhniki, Akad. Nauk SSSR,Vsesoyuz. Inst. Nauchn. i Tekhn. [9] Tulovskiˇi, V. N. ; Shubin, M. A. The asymptotic distribution of the eigenvaluesof pseudodifferential operators in R n L functions on coverings ofpseudoconvex manifolds. Geom. Funct. Anal. 8 (1998), no. 3, 552–585.[27] M. Gromov, M. Shubin, von Neumann spectra near zero. Geom. Funct. Anal.1 (1991), no. 4, 375–404. [28] M. Gromov, M. Shubin, The Riemann-Roch theorem for elliptic operators. I.M. Gel’fand Seminar, 211–241, Adv. Soviet Math., 16, Part 1, Amer. Math.Soc., Providence, RI, 1993.[29] M. Gromov, M. Shubin, The Riemann-Roch theorem for elliptic operators andsolvability of elliptic equations with additional conditions on compact subsets.Journes ”quations aux Drives Partielles” (Saint-Jean-de-Monts, 1993), Exp.No. XVIII, 13 pp., ´Ecole Polytech., Palaiseau, 1993.[30] M. Gromov, M. Shubin, The Riemann-Roch theorem for elliptic operators andsolvability of elliptic equations with additional conditions on compact subsets.Invent. Math. 117 (1994), no. 1, 165–180.[31] T. Kappeler, P. Perry, M. Shubin, P. Topalov, The Miura map on the line. Int.Math. Res. Not. 2005, no. 50, 3091–3133.[32] T. Kappeler, P. Perry, M. Shubin, P. Topalov, Solutions of mKdV in classesof functions unbounded at infinity. J. Geom. Anal. 18 (2008), no. 2, 443–477.[33] V. Kondratiev, V. Maz’ya, M. Shubin, Gauge optimization and spectral prop-erties of magnetic Schrdinger operators. Comm. Partial Differential Equations34 (2009), no. 10-12, 1127–1146.[34] V. Kondratiev, V. Maz’ya, M. Shubin, Discreteness of spectrum and strict pos-itivity criteria for magnetic Schrdinger operators. Comm. Partial DifferentialEquations 29 (2004), no. 3-4, 489–521.[35] V. Kondratiev, M. Shubin, Discreteness of spectrum for the Schrdinger oper-ators on manifolds of bounded geometry. The Maz’ya anniversary collection,Vol. 2 (Rostock, 1998), 185–226, Oper. Theory Adv. Appl., 110, Birkhuser,Basel, 1999.[36] V. Kondratiev, M. Shubin, Discreteness of spectrum for the magneticSchr¨odinger operators. Comm. Partial Differential Equations 27 (2002), no.3-4, 477–525.[37] V. Maz’ya, M. Shubin, Discreteness of spectrum and positivity criteria forSchr¨odinger operators. Ann. of Math. (2) 162 (2005), no. 2, 919–942.[38] V. Maz’ya, Vladimir, M. Shubin, Can one see the fundamental frequency of adrum? Lett. Math. Phys. 74 (2005), no. 2, 135–151.[39] M. Shubin, Spectral theory of elliptic operators on noncompact manifolds,M´ethodes semi-classiques, Vol. (Nantes, 1991). Ast´erisque No. 207 (1992),5, 35–108.[40] M. Shubin, Discrete magnetic Laplacian. Comm. Math. Phys. (1994), no.2, 259–275.[41] M. Shubin, Spectral theory of the Schr¨odinger operators on non-compact man-ifolds: qualitative results. Spectral theory and geometry (Edinburgh, 1998),226–283, London Math. Soc. Lecture Note Ser., 273, Cambridge Univ. Press,Cambridge, 1999.[42] M. Shubin, L Riemann-Roch theorem for elliptic operators. Geom. Funct.Anal.5