@article{RM_2006_61_5_a1,
author = {V. A. Sadovnichii and V. E. Podolskii},
title = {Traces of operators},
journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
pages = {885--953},
year = {2006},
volume = {61},
number = {5},
language = {en},
url = {http://geodesic.mathdoc.fr/item/RM_2006_61_5_a1/}
}
V. A. Sadovnichii; V. E. Podolskii. Traces of operators. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 61 (2006) no. 5, pp. 885-953. http://geodesic.mathdoc.fr/item/RM_2006_61_5_a1/
[1] I. Ts. Gokhberg, M. G. Krein, Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve, Nauka, M., 1965 | MR | Zbl
[2] V. B. Lidskii, “Nesamosopryazhennye operatory, imeyuschie sled”, Dokl. AN SSSR, 125:3 (1959), 485–487 | MR | Zbl
[3] I. M. Lifshits, “Ob odnoi zadache teorii vozmuschenii, svyazannoi s kvantovoi statistikoi”, UMN, 7:1 (1952), 171–180 | MR | Zbl
[4] M. G. Krein, “O formule sledov v teorii vozmuschenii”, Matem. sb., 33(75):3 (1953), 597–626 | MR | Zbl
[5] D. R. Yafaev, Matematicheskaya teoriya rasseyaniya, Izd-vo S.-Peterburgskogo un-ta, SPb., 1994 | MR | Zbl
[6] M. Sh. Birman, D. R. Yafaev, “Funktsiya spektralnogo sdviga. Raboty M. G. Kreina i ikh dalneishee razvitie”, Algebra i analiz, 4:5 (1992), 1–44 | MR | Zbl
[7] I. M. Gelfand, B. M. Levitan, “Ob odnom prostom tozhdestve dlya sobstvennykh znachenii differentsialnogo operatora vtorogo poryadka”, Dokl. AN SSSR, 88:4 (1953), 593–596 | MR | Zbl
[8] L. A. Dikii, “Ob odnoi formule Gelfanda–Levitana”, UMN, 8:2 (1953), 119–123 | MR | Zbl
[9] R. C. Gilbert, V. A. Kramer, “Trace formulas for powers of a Sturm–Liouville operator”, Canad. J. Math., 16:4 (1964), 412–422 | MR | Zbl
[10] M. G. Gasymov, “O summe raznostei sobstvennykh znachenii dvukh samosopryazhennykh operatorov”, Dokl. AN SSSR, 150:6 (1963), 1202–1205 | MR | Zbl
[11] V. B. Lidskii, V. A. Sadovnichii, “Regulyarizovannye summy kornei odnogo klassa tselykh funktsii”, Funkts. analiz i ego prilozh., 1:2 (1967), 52–59 | MR | Zbl
[12] L. D. Faddeev, “O vyrazhenii dlya sleda raznosti dvukh singulyarnykh differentsialnykh operatorov tipa Shturma–Liuvillya”, Dokl. AN SSSR, 115:5 (1957), 878–881 | MR | Zbl
[13] V. S. Buslaev, L. D. Faddeev, “O formulakh sledov dlya differentsialnogo singulyarnogo operatora Shturma–Liuvillya”, Dokl. AN SSSR, 132:1 (1960), 13–16 | MR | Zbl
[14] V. E. Zakharov, L. D. Faddeev, “Uravnenie Kortevega–de Frisa – vpolne integriruemaya gamiltonova sistema”, Funkts. analiz i ego prilozh., 5:4 (1971), 18–27 | MR | Zbl
[15] V. A. Sadovnichii, V. V. Dubrovskii, “Ob odnoi abstraktnoi teoreme teorii vozmuschenii, o formulakh regulyarizovannykh sledov i o $\zeta$-funktsii operatorov”, Differents. uravneniya, 13:7 (1977), 1264–1271 | MR | Zbl
[16] V. A. Sadovnichii, V. V. Dubrovskii, “O nekotorykh sootnosheniyakh dlya sobstvennykh chisel diskretnykh operatorov. Formuly sledov dlya differentsialnykh operatorov v chastnykh proizvodnykh”, Differents. uravneniya, 13:11 (1977), 2033–2042 | MR | Zbl
[17] V. A. Sadovnichii, V. E. Podolskii, “Sledy operatorov s otnositelno kompaktnym vozmuscheniem”, Matem. sb., 193:2 (2002), 129–152 | MR | Zbl
[18] V. A. Sadovnichii, V. V. Dubrovskii, “O klassicheskoi formule pervogo regulyarizovannogo sleda operatora Laplasa s nechetnym potentsialom na sfere”, Tr. sem. im. I. G. Petrovskogo, 19, 1996, 37–72 | MR | Zbl
[19] V. E. Podolskii, “Formula regulyarizovannogo sleda operatora Laplasa–Beltrami s nechetnym potentsialom na sfere $S^2$”, Matem. zametki, 56:1 (1994), 71–77 | MR | Zbl
[20] V. E. Podol'skii, “On the summability of regularized sums of eigenvalues of the Laplace–Beltrami operator with potential on symmetric spaces of rank one”, Russian J. Math. Phys., 4:1 (1996), 123–130 | MR | Zbl
[21] P. D. Lax, “Trace formulas for the Schrödinger operator”, Comm. Pure Appl. Math., 47:4 (1994), 503–512 | DOI | MR | Zbl
[22] F. Gesztesy, H. Holden, B. Simon, Z. Zhao, “A trace formula for multidimensional Schrödinger operators”, J. Funct. Anal., 141:2 (1996), 449–465 | DOI | MR | Zbl
[23] V. A. Lyubishkin, V. E. Podolskii, “O summiruemosti regulyarizovannykh sledov differentsialnykh operatorov”, Matem. zametki, 54:2 (1993), 33–38 | MR | Zbl
[24] F. Gesztesy, H. Holden, B. Simon, Z. Zhao, “Trace formulae and inverse spectral theory for Schrödinger operators”, Bull. Amer. Math. Soc. (N.S.), 29:2 (1993), 250–255 | DOI | MR | Zbl
[25] A. B. Venkov, “Spektralnaya teoriya avtomorfnykh funktsii, $\zeta$-funktsiya Selberga i nekotorye problemy analiticheskoi teorii chisel i matematicheskoi fiziki”, UMN, 34:3 (1979), 69–135 | MR | Zbl
[26] I. M. Gelfand, “O tozhdestvakh dlya sobstvennykh znachenii differentsialnogo operatora vtorogo poryadka”, UMN, 11:1 (1956), 191–198 | MR | Zbl
[27] V. A. Marchenko, Operatory Shturma–Liuvillya i ikh prilozheniya, Naukova dumka, Kiev, 1977 | MR | Zbl
[28] V. V. Dubrovskii, “Regulyarizovannyi sled operatora Shturma–Liuvillya”, Differents. uravneniya, 16:6 (1980), 1127–1129 | MR | Zbl
[29] V. A. Vinokurov, V. A. Sadovnichii, “Sobstvennoe znachenie i sled operatora Shturma–Liuvillya kak differentsiruemye funktsii summiruemogo potentsiala”, Dokl. RAN, 365:3 (1999), 295–297 | MR | Zbl
[30] V. A. Vinokurov, V. A. Sadovnichii, “Asimptotika sobstvennykh znachenii i sobstvennykh funktsii i formula sleda dlya potentsiala, soderzhaschego $\delta$-funktsii”, Dokl. RAN, 376:4 (2001), 445–448 | MR | Zbl
[31] A. M. Savchuk, “Regulyarizovannyi sled pervogo poryadka operatora Shturma–Liuvillya s $\delta$-potentsialom”, UMN, 55:6 (2000), 155–156 | MR | Zbl
[32] A. M. Savchuk, A. A. Shkalikov, “Formula sleda dlya operatorov Shturma–Liuvillya s singulyarnymi potentsialami”, Matem. zametki, 69:3 (2001), 427–442 | MR | Zbl
[33] L. A. Dikii, “Dzeta-funktsiya obyknovennogo differentsialnogo uravneniya na konechnom otrezke”, Izv. AN SSSR. Ser. matem., 19:4 (1955), 187–200 | MR | Zbl
[34] L. A. Dikii, “Formuly sledov dlya differentsialnykh operatorov Shturma–Liuvillya”, UMN, 13:3 (1958), 111–143 | MR | Zbl
[35] B. M. Levitan, “Vychislenie regulyarizovannogo sleda dlya operatora Shturma–Liuvillya”, UMN, 19:1 (1964), 161–165 | MR | Zbl
[36] C. J. A. Halberg, Jr., V. A. Kramer, “A generalization of the trace concept”, Duke Math. J., 27:4 (1960), 607–617 | DOI | MR | Zbl
[37] M. G. Gasymov, B. M. Levitan, “O summe raznostei sobstvennykh znachenii dvukh singulyarnykh operatorov Shturma–Liuvillya”, Dokl. AN SSSR, 151:5 (1963), 1014–1017 | MR | Zbl
[38] S. I. Grinberg, “Formula sledov dlya uravneniya Shredingera v konechnoi oblasti”, Matem. zametki, 1:4 (1967), 451–460 | MR | Zbl
[39] R. F. Shevchenko, “Regulyarizatsiya sleda obyknovennogo differentsialnogo operatora”, Vestn. MGU. Ser. 1. Matem., mekh., 1965, no. 6, 28–36 | MR | Zbl
[40] A. G. Kostyuchenko, O nekotorykh spektralnykh svoistvakh differentsialnykh operatorov, Dis. ... dokt. fiz.-matem. nauk, MGU, M., 1966
[41] V. A. Sadovnichii, “O slede raznosti dvukh obyknovennykh differentsialnykh operatorov vysshikh poryadkov”, Differents. uravneniya, 2:12 (1966), 1611–1624 | MR | Zbl
[42] V. A. Sadovnichii, “O sledakh obyknovennykh differentsialnykh operatorov vysshikh poryadkov”, Matem. sb., 72:2 (1967), 293–317 | MR | Zbl
[43] V. A. Sadovnichii, “O tozhdestvakh dlya sobstvennykh znachenii sistemy Diraka i nekotorykh drugikh sistem vysshego poryadka”, Vestn. MGU. Ser. 1. Matem., mekh., 1967, no. 3, 37–47 | MR | Zbl
[44] E. Abdukadyrov, “Vychislenie regulyarizovannogo sleda dlya sistemy Diraka”, Vestn. MGU. Ser. 1. Matem., mekh., 1967, no. 4, 17–24 | MR | Zbl
[45] A. Zommerfeld, Stroenie atoma i spektry, t. 1, Gostekhizdat, M., 1956
[46] M. A. Naimark, Lineinye differentsialnye operatory, Nauka, M., 1969 | MR | Zbl
[47] V. B. Lidskii, V. A. Sadovnichii, “Formuly sledov v sluchae uravneniya Orra–Zommerfelda”, Izv. AN SSSR. Ser. matem., 32:3 (1968), 633–648 | MR | Zbl
[48] V. A. Sadovnichii, “Regulyarizovannye summy polutselykh stepenei operatora Shturma–Liuvillya”, Matem. zametki, 14:2 (1973), 279–290 | MR | Zbl
[49] V. A. Sadovnichii, “O nekotorykh tozhdestvakh dlya sobstvennykh chisel singulyarnykh obyknovennykh differentsialnykh operatorov. Sootnosheniya dlya nulei funktsii Besselya”, Vestn. MGU. Ser. 1. Matem., mekh., 1971, no. 3, 77–86 | MR | Zbl
[50] P. L. Kapitsa, “Vychislenie summ otritsatelnykh chetnykh stepenei kornei besselevykh funktsii”, Dokl. AN SSSR, 77:4 (1951), 561–564 | MR | Zbl
[51] N. N. Meiman, “O rekurrentnykh formulakh dlya stepennykh summ nulei besselevykh funktsii”, Dokl. AN SSSR, 108:2 (1956), 190–193 | MR | Zbl
[52] V. A. Sadovnichii, “O sledakh s vesom i ob asimptotike spektralnoi funktsii”, Differents. uravneniya, 10:10 (1974), 1808–1818 | MR | Zbl
[53] A. A. Stakun, “O svoistvakh operatora Shredingera s kompleksnoznachnym potentsialom”, Differents. uravneniya, 32:3 (1996), 423–425 | MR | Zbl
[54] A. S. Pechentsov, “Regulyarizovannye sledy differentsialnykh operatorov: metod Lidskogo–Sadovnichego”, Differents. uravneniya, 35:4 (1999), 490–497 | MR | Zbl
[55] A. S. Pechentsov, “Sledy odnogo klassa singulyarnykh differentsialnykh operatorov: metod Lidskogo–Sadovnichego”, Vestn. MGU. Ser. 1. Matem., mekh., 1999, no. 5, 35–42 | MR | Zbl
[56] Yu. Belabbasi, “Regulyarizovannye sledy mnogotochechnoi zadachi dlya obyknovennykh differentsialnykh operatorov vysshikh poryadkov”, Differents. uravneniya, 19:6 (1983), 938–944 | MR | Zbl
[57] M. Martinovich, “Dzeta-funktsiya i formuly sledov dlya odnoi kraevoi zadachi s funktsionalno-differentsialnym uravneniem”, Differents. uravneniya, 18:3 (1982), 537–540 | MR | Zbl
[58] Z. Kadelburg, “Ob asimptotike «spektralnykh funktsii» dvukh nesamosopryazhennykh kraevykh zadach”, Matem. zametki, 30:5 (1981), 659–667 | MR | Zbl
[59] Z. Kadelburg, “Ob asimptotike spektralnoi funktsii dlya obyknovennykh differentsialnykh operatorov, zadannykh dvuchlennoi operatsiei na otrezke”, Differents. uravneniya, 18:9 (1982), 1475–1480 | MR | Zbl
[60] Z. Kadelburg, M. Martinovich, “Spektralnaya funktsiya odnogo funktsionalno-differentsialnogo operatora vtorogo poryadka”, Differents. uravneniya, 25:11 (1989), 1882–1888 | MR | Zbl
[61] S. I. Mitrokhin, “O nekotorykh spektralnykh svoistvakh differentsialnykh operatorov vtorogo poryadka s razryvnoi polozhitelnoi vesovoi funktsiei”, Dokl. RAN, 356:1 (1997), 13–15 | MR | Zbl
[62] M. Pikula, “O regulyarizovannykh sledakh differentsialnogo operatora tipa Shturma–Liuvillya s zapazdyvayuschim argumentom”, Differents. uravneniya, 26:1 (1990), 103–109 | MR | Zbl
[63] V. A. Sadovnichii, V. A. Lyubishkin, Yu. Belabbasi, “O nulyakh tselykh funktsii odnogo klassa”, Tr. sem. im. I. G. Petrovskogo, 8, 1982, 211–217 | MR | Zbl
[64] V. A. Sadovnichii, V. A. Lyubishkin, “Regulyarizovannye summy kornei odnogo klassa tselykh funktsii eksponentsialnogo tipa”, Dokl. AN SSSR, 256:4 (1981), 794–798 | MR | Zbl
[65] S. A. Salekh, “Regulyarizovannye sledy mnogotochechnykh zadach”, Vestn. MGU. Ser. 1. Matem., mekh., 1985, no. 5, 64–66 | MR | Zbl
[66] G. S. Tipenko, “Regulyarizovannye summy vychetov dlya odnogo klassa tselykh funktsii eksponentsialnogo tipa”, Vestn. MGU. Ser. 1. Matem., mekh., 1985, no. 1, 93–96 | MR | Zbl
[67] Yu. V. Matveev, “Nekotorye svoistva sobstvennykh funktsii differentsialno-funktsionalnogo operatora”, Vestn. MGU. Ser. vychisl. matem. i kibern., 1984, no. 3, 54–59 | MR | Zbl
[68] V. S. Buslaev, “Formuly sledov dlya operatora Shrëdingera v trekhmernom prostranstve”, Dokl. AN SSSR, 143:5 (1962), 1067–1070 | MR | Zbl
[69] V. S. Buslaev, “Formuly sledov i nekotorye asimptoticheskie otsenki yadra rezolventy dlya operatora Shrëdingera v trekhmernom prostranstve”, Spektralnaya teoriya i volnovye protsessy, Problemy matem. fiziki, 1, no. 1, eds. M. Sh. Birman, Izd-vo LGU, L., 1966, 82–101 | MR | Zbl
[70] V. S. Buslaev, “Rasseyannye ploskie volny, spektralnye asimptotiki i formuly sleda vo vneshnikh zadachakh”, Dokl. AN SSSR, 197:5 (1971), 999–1002 | MR | Zbl
[71] A. Rybkin, “On a trace formula of the Buslaev–Faddeev type for a long-range potential”, J. Math. Phys., 40:3 (1999), 1334–1343 | DOI | MR | Zbl
[72] S. M. Belov, A. V. Rybkin, “Higher order trace formulas of the Buslaev–Faddeev-type for the half-line Schrödinger operator with long-range potential”, J. Math. Phys., 44:7 (2003), 2748–2761 | DOI | MR | Zbl
[73] F. A. Berezin, “Formula sledov dlya mnogochastichnogo uravneniya Shredingera”, Dokl. AN SSSR, 157:5 (1964), 1069–1072
[74] V. S. Buslaev, “Formula sleda i singulyarnosti matritsy rasseyaniya dlya sistemy trekh odnomernykh chastits. Tretii gruppovoi integral”, TMF, 16:2 (1973), 247–259 | MR
[75] S. P. Merkurev, “Ob $S$-matrichnoi regulyarizatsii formuly sleda dlya sistemy trekh chastits”, Zapiski nauch. sem. LOMI, 63, 1976, 95–131 | MR | Zbl
[76] E. Beth, G. E. Uhlenbeck, “The quantum theory of the non-ideal gas. II: Behaviour at low temperatures”, Physica, 4:10 (1937), 915–924 | DOI | Zbl
[77] D. R. Yafaev, “O formule sleda v mnogokanalnoi modeli Fridrikhsa”, Kraevye zadachi matematicheskoi fiziki. 10, Tr. MIAN, 147, ed. O. A. Ladyzhenskaya, Nauka, L., 1980, 194–201 | MR | Zbl
[78] S. A. Stepin, “Spektr rezonansov i formula sleda v zadache potentsialnogo rasseyaniya”, Funkts. analiz i ego prilozh., 38:3 (2004), 79–89 | MR | Zbl
[79] A. V. Rybkin, “Formula sledov dlya rezonansov”, TMF, 56:3 (1983), 439–447 | MR | Zbl
[80] H. Hochstadt, “On the determination of a Hill's equation from its spectrum”, Arch. Rational Mech. Anal., 19:5 (1965), 353–362 | DOI | MR | Zbl
[81] I. M. Gelfand, L. A. Dikii, “Asimptotika rezolventy shturm-liuvillevskikh uravnenii i algebra uravnenii Kortevega–de Friza”, UMN, 30:5 (1975), 67–100 | MR | Zbl
[82] B. M. Levitan, “Esche odin sposob vychisleniya plotnostei integralov dvizheniya dlya uravneniya Kortevega–de Friza”, Matem. zametki, 22:1 (1977), 129–135 | MR | Zbl
[83] M. Ablovits, Kh. Sigur, Solitony i metod obratnoi zadachi, Mir, M., 1987 | MR | MR | Zbl
[84] F. Gesztesy, H. Holden, “Trace formulas and conservation laws for nonlinear evolution equations”, Rev. Math. Phys., 6:1 (1994), 51–95 | DOI | MR | Zbl
[85] A. Rybkin, “KdV invariants and Herglotz functions”, Differential Integral Equations, 14:4 (2001), 493–512 | MR | Zbl
[86] P. Deift, R. Killip, “On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials”, Comm. Math. Phys., 203:2 (1999), 341–347 | DOI | MR | Zbl
[87] H. P. McKean, P. van Moerbeke, “The spectrum of Hill's equation”, Invent. Math., 30:3 (1975), 217–274 | DOI | MR | Zbl
[88] H. Flaschka, “On the inverse problem for Hill's operator”, Arch. Rational Mech. Anal., 59:4 (1975), 293–309 | DOI | MR | Zbl
[89] S. Kotani, M. Krishna, “Almost periodicity of some random potentials”, J. Funct. Anal., 78:2 (1988), 390–405 | DOI | MR | Zbl
[90] W. Craig, “The trace formula for Schrödinger operators on the line”, Comm. Math. Phys., 126:2 (1989), 379–407 | DOI | MR | Zbl
[91] B. M. Levitan, “Regulyarizovannye sledy i usloviya gladkoi periodichnosti potentsiala uravneniya Shturma–Liuvillya”, Sib. matem. zhurn., 22:2 (1981), 137–148 | MR | Zbl
[92] G. Sh. Guseinov, B. M. Levitan, “O formulakh sledov dlya operatorov Shturma–Liuvillya”, Vestn. MGU. Ser. 1. Matem., mekh., 1978, no. 1, 40–49 | MR | Zbl
[93] V. A. Sadovnichii, V. A. Lyubishkin, “Konechnomernye vozmuscheniya diskretnykh operatorov i formuly sledov”, Funkts. analiz i ego prilozh., 20:3 (1986), 55–65 | MR | Zbl
[94] V. V. Dubrovskii, A. S. Pechentsov, “Regulyarizovannye sledy ellipticheskikh operatorov vysshikh poryadkov”, Differents. uravneniya, 29:1 (1993), 50–53 | MR | Zbl
[95] V. A. Lyubishkin, I. D. Tsopanov, “Regulyarizovannye sledy integro-differentsialnykh operatorov”, Matem. zametki, 43:6 (1988), 786–793 | MR | Zbl
[96] V. V. Dubrovskii, “Formuly regulyarizovannykh sledov operatorov s kompaktnoi rezolventoi”, Differents. uravneniya, 26:12 (1990), 2046–2051 | MR | Zbl
[97] V. V. Dubrovskii, “Abstraktnye formuly regulyarizovannykh sledov ellipticheskikh gladkikh differentsialnykh operatorov, zadannykh na kompaktnykh mnogoobraziyakh”, Differents. uravneniya, 27:12 (1991), 2164–2166 | MR | Zbl
[98] V. V. Dubrovskii, E. A. Puzankova, “Otsenka raznosti spektralnykh funktsii i formuly regulyarizovannykh sledov stepeni operatora Laplasa, zadannogo na treugolnike ili kvadrate, v $L_p$, $1\leqslant p\leqslant 2$”, Differents. uravneniya, 35:4 (1999), 552–555 | MR | Zbl
[99] V. A. Lyubishkin, “O formulakh sledov Gelfanda–Levitana i Kreina”, Matem. sb., 182:12 (1991), 1786–1795 | MR | Zbl
[100] V. V. Dubrovskii, A. I. Sedov, “Otsenka raznosti spektralnykh funktsii operatorov tipa Lezhandra”, Fundam. i prikl. matem., 6:4 (2000), 1075–1082 | MR | Zbl
[101] V. A. Sadovnichii, V. V. Dubrovskii, N. Yu. Sochenko, “Regulyarizovannye sledy nesamosopryazhennykh diskretnykh operatorov s neyadernoi rezolventoi”, Dokl. RAN, 370:1 (2000), 24–26 | MR | Zbl
[102] A. G. Baskakov, “Formuly regulyarizovannykh sledov dlya stepenei vozmuschennykh spektralnykh operatorov”, Izv. vuzov. Ser. matem., 1985, no. 8, 68–71 | MR | Zbl
[103] S. V. Sidorenko, “O formulakh regulyarizovannykh sledov”, UMN, 54:5 (1999), 173–174 | MR | Zbl
[104] M. Dostanić, “Trace formulas of Gel'fand–Levitan type”, Publ. Inst. Math. (Beograd) (N.S.), 55(69) (1994), 51–63 | MR | Zbl
[105] V. A. Sadovnichii, V. E. Podolskii, “Regulyarizovannyi sled ogranichennogo vozmuscheniya operatora s yadernoi rezolventoi”, Differents. uravneniya, 35:4 (1999), 556–564 | MR | Zbl
[106] V. A. Sadovnichii, S. V. Konyagin, V. E. Podolskii, “Regulyarizovannyi sled operatora s yadernoi rezolventoi, vozmuschennogo ogranichennym”, Dokl. RAN, 373:1 (2000), 26–28 | MR | Zbl
[107] V. V. Dubrovskii, A. N. Tipko, Z. S. Chekashkina, “Regulyarizovannye sledy unitarnykh operatorov”, UMN, 56:6 (2001), 145–146 | MR | Zbl
[108] Kh. Kh. Murtazin, Z. Yu. Fazullin, “O formulakh sledov dlya neyadernykh vozmuschenii”, Dokl. RAN, 368:4 (1999), 442–444 | MR | Zbl
[109] Kh. Kh. Murtazin, Z. Yu. Fazullin, “Neyadernye vozmuscheniya diskretnykh operatorov i formuly sledov”, Matem. sb., 196:12 (2005), 123–156 | MR
[110] T. G. Amangildin, “Regulyarizovannyi sled operatora Shturma–Liuvillya”, Differents. uravneniya, 25:8 (1989), 1439–1441 | MR | Zbl
[111] G. V. Kozlov, V. A. Lyubishkin, “Regulyarizovannye sledy singulyarnykh differentsialnykh operatorov”, Vestn. MGU. Ser. 1. Matem., mekh., 1993, no. 4, 6–11 | MR | Zbl
[112] G. V. Kozlov, V. A. Lyubishkin, “Regulyarizovannye sledy vysshikh poryadkov dlya garmonicheskogo ostsillyatora”, Differents. uravneniya, 29:1 (1993), 61–63 | MR | Zbl
[113] A. I. Kozko, A. S. Pechentsov, “Spektralnaya funktsiya i regulyarizovannye sledy singulyarnykh differentsialnykh operatorov vysshikh poryadkov”, Dokl. RAN, 401:2 (2005), 160–162 | MR
[114] V. A. Lyubishkin, “Regulyarizovannye sledy operatora Shturma–Liuvillya na poluosi v sluchae neogranichenno ubyvayuschego potentsiala”, Differents. uravneniya, 18:2 (1982), 345–346 | MR | Zbl
[115] V. V. Dubrovskii, A. I. Sedov, “Asimptotika sobstvennykh znachenii singulyarnogo differentsialnogo operatora tipa Yakobi”, Dokl. RAN, 353:3 (1997), 295–299 | MR | Zbl
[116] V. E. Podolskii, “O vosstanovlenii operatora Shturma–Liuvillya po ego vesovoi $\zeta$-funktsii”, Dokl. AN SSSR, 313:3 (1990), 559–562 | MR | Zbl
[117] E. V. Aleksandrova, “Formula sleda dlya uravnenii kolebanii pryamolineinogo truboprovoda”, Differents. uravneniya, 33:1 (1997), 12–18 | MR | Zbl
[118] V. B. Lidskii, “O summiruemosti ryadov po glavnym vektoram nesamosopryazhennykh operatorov”, Tr. MMO, 11, 1962, 3–35 | MR | Zbl
[119] V. E. Podolskii, “Summirovanie po Abelyu regulyarizovannykh sledov”, Vestn. MGU. Ser. 1. Matem., mekh., 1999, no. 5, 42–48 | MR | Zbl
[120] E. M. Santanzhelo, “Vychislenie energii Kazimira s pomoschyu spektralnykh funktsii”, TMF, 131:1 (2002), 98–117 | MR | Zbl
[121] E. Elizalde, “Zeta functions: formulas and applications”, J. Comput. Appl. Math., 118:1–2 (2000), 125–142 | DOI | MR | Zbl
[122] E. V. Aleksandrova, O. V. Bochkareva, V. E. Podolskii, “Summirovanie regulyarizovannykh sledov singulyarnogo operatora Shturma–Liuvillya”, Differents. uravneniya, 33:3 (1997), 291–295 | MR | Zbl
[123] E. V. Aleksandrova, Formula sleda garmonicheskogo ostsillyatora s nefinitnym vozmuscheniem, Dep. v VINITI RAN 18.01.1997 No 129-B97
[124] A. Pushnitski, I. Sorrell, “High energy asymptotics and trace formulas for the perturbed harmonic oscillator”, Ann. Henri Poincaré, 7:2 (2006), 381–396 | DOI | MR | Zbl
[125] M. Hitrik, I. Polterovich, “Regularized traces and Taylor expansions for the heat semigroup”, J. London Math. Soc. (2), 68:2 (2003), 402–418 | DOI | MR | Zbl
[126] V. E. Podolskii, “O slede vozmuschennoi operatornoi polugruppy”, Matem. zametki, 75:3 (2004), 462–465 | MR | Zbl
[127] N. Khart, Geometricheskoe kvantovanie v deistvii: Prilozheniya garmonicheskogo analiza v kvantovoi statisticheskoi mekhanike i kvantovoi teorii polya, Mir, M., 1985 | MR | MR | Zbl
[128] V. V. Dubrovskii, “O regulyarizovannykh sledakh differentsialnykh operatorov v chastnykh proizvodnykh”, Tr. sem. im. I. G. Petrovskogo, 9, 1983, 40–44 | MR | Zbl
[129] A. N. Bobrov, V. E. Podolskii, “Skhodimost regulyarizovannykh sledov stepeni operatora Laplasa–Beltrami s potentsialom na sfere $S^n$”, Matem. sb., 190:10 (1999), 3–16 | MR | Zbl
[130] A. N. Bobrov, “Regulyarizovannye sledy vysshikh poryadkov operatora Laplasa s potentsialom na simmetricheskikh prostranstvakh ranga 1”, Differents. uravneniya, 33:6 (1997), 800–804 | MR | Zbl
[131] V. A. Sadovnichii, Z. Yu. Fazullin, “Asimptotika sobstvennykh chisel i formula sleda vozmuscheniya operatora Laplasa na sfere $S^2$”, Matem. zametki, 77:3 (2005), 434–448 | MR | Zbl
[132] V. A. Lyubishkin, “Regulyarizovannyi sled dvumernogo ostsillyatora”, Matem. zametki, 53:3 (1993), 156–158 | MR | Zbl
[133] Z. Yu. Fazullin, Kh. Kh. Murtazin, “Klassicheskaya formula regulyarizovannogo sleda mnogomernogo garmonicheskogo ostsillyatora”, Tr. sem. im. I. G. Petrovskogo, 21, 2001, 298–339 | MR | Zbl
[134] Kh. Kh. Murtazin, Z. Yu. Fazullin, “Spektr i formula sledov dlya dvumernogo operatora Shredingera v odnorodnom magnitnom pole”, Dokl. RAN, 390:6 (2003), 743–745 | MR | Zbl
[135] E. Korotyaev, A. Pushnitski, “A trace formula and high-energy spectral asymptotics for the perturbed Landau Hamiltonian”, J. Funct. Anal., 217:1 (2004), 221–248 | DOI | MR | Zbl
[136] M. R. Dostanić, “Spectral properties of the operator of Riesz potential type”, Proc. Amer. Math. Soc., 126:8 (1998), 2291–2297 | DOI | MR | Zbl
[137] Yu. V. Kogan, “Formuly sledov dlya psevdodifferentsialnykh operatorov na okruzhnosti”, Vestn. MGU. Ser. 1. Matem., mekh., 1980, no. 6, 7–11 | MR | Zbl
[138] V. E. Podolskii, “Regulyarizovannye sledy nekotorykh psevdodifferentsialnykh operatorov na okruzhnosti”, Vestn. MGU. Ser. 1. Matem., mekh., 1988, no. 6, 11–14 | MR | Zbl
[139] A. N. Bobrov, “Spektralnye svoistva nekotorykh dvumernykh kraevykh zadach”, Vestn. MGU. Ser. 1. Matem., mekh., 2001, no. 1, 3–6 | MR | Zbl
[140] I. M. Lifshits, “O vyrozhdennykh regulyarnykh vozmuscheniyakh. I: Diskretnyi spektr”, ZhETF, 17:11 (1947), 1017–1025
[141] I. M. Lifshits, “O vyrozhdennykh regulyarnykh vozmuscheniyakh. II: Kvazinepreryvnyi i nepreryvnyi spektry”, ZhETF, 17:12 (1947), 1076–1089
[142] M. G. Krein, “Ob opredelitelyakh vozmuscheniya i formule sledov dlya unitarnykh i samosopryazhennykh operatorov”, Dokl. AN SSSR, 144:2 (1962), 268–271 | MR | Zbl
[143] M. Sh. Birman, M. G. Krein, “K teorii volnovykh operatorov i operatorov rasseyaniya”, Dokl. AN SSSR, 144:3 (1962), 475–478 | MR | Zbl
[144] V. A. Yavryan, “O funktsii spektralnogo sdviga dlya operatorov Shturma–Liuvillya”, Dokl. AN ArmSSR, 38:4 (1964), 193–198 | MR | Zbl
[145] L. S. Koplienko, “O formule sledov dlya vozmuschenii neyadernogo tipa”, Sib. matem. zhurn., 25:5 (1984), 62–71 | MR | Zbl
[146] H. Neidhardt, “Spectral shift function and Hilbert–Schmidt perturbation: extensions of some work of L. S. Koplienko”, Math. Nachr., 138 (1988), 7–25 | DOI | MR | Zbl
[147] M. Dostanić, “Trace formula for nonnuclear perturbations of selfadjoint operators”, Publ. Inst. Math. (Beograd) (N.S.), 54(68) (1993), 71–79 | MR | Zbl
[148] A. V. Rybkin, “Funktsiya spektralnogo sdviga, kharakteristicheskaya funktsiya szhatiya i obobschennyi integral”, Matem. sb., 185:10 (1994), 91–144 | MR | Zbl
[149] A. Poltoratskii, “Spektralnyi sdvig Kreina i vozmuscheniya spektrov ranga odin”, Algebra i analiz, 10:5 (1998), 143–183 | MR | Zbl
[150] A. B. Pushnitskii, “Predstavlenie dlya funktsii spektralnogo sdviga v sluchae znakoopredelennykh vozmuschenii”, Algebra i analiz, 9:6 (1997), 197–213 | MR | Zbl
[151] A. B. Pushnitskii, “Integralnye otsenki dlya funktsii spektralnogo sdviga”, Algebra i analiz, 10:6 (1998), 198–233 | MR | Zbl
[152] E. Korotyaev, A. Pushnitski, “Trace formulae and high energy asymptotics for the Stark operator”, Comm. Partial Differential Equations, 28:3–4 (2003), 817–842 | DOI | MR | Zbl
[153] W. Müller, “Relative zeta functions, relative determinants and scattering theory”, Comm. Math. Phys., 192:2 (1998), 309–347 | DOI | MR | Zbl
[154] F. Gesztesy, B. Simon, “The xi function”, Acta Math., 176:1 (1996), 49–71 | DOI | MR | Zbl
[155] A. Rybkin, “Necessary and sufficient conditions for absolute summability of the trace formulas for certain one dimensional Schrödinger operators”, Proc. Amer. Math. Soc., 131:1 (2003), 219–229 | DOI | MR | Zbl
[156] V. A. Sadovnichii, V. E. Podolskii, “Ob obobschennoi funktsii spektralnogo sdviga i svyazi formul sleda Kreina i Gelfanda–Levitana”, Dokl. RAN, 402:3 (2005), 311–312 | MR
[157] H. Weyl, “Ramifications, old and new, of the eigenvalue problem”, Bull. Amer. Math. Soc., 56 (1950), 115–139 | DOI | MR | Zbl
[158] R. T. Sili, “Stepeni ellipticheskogo operatora”, Matematika, 12:1 (1968), 96–112
[159] M. A. Shubin, Psevdodifferentsialnye operatory i spektralnaya teoriya, Nauka, M., 1978 | MR | Zbl
[160] M. F. Atiyah, V. K. Patodi, I. M. Singer, “Spectral asymmetry and Riemannian geometry, III”, Math. Proc. Cambridge Philos. Soc., 79:1 (1976), 71–99 | DOI | MR | Zbl
[161] P. B. Gilkey, “The residue of the global $\eta$-function at the origin”, Advin. in Math., 40:3 (1981), 290–307 | DOI | MR | Zbl
[162] M. Wodzicki, “Noncommutative residue. I: Fundamentals”, $K$-theory, aritmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math., 1289, Springer, Berlin, 1987, 320–399 | MR | Zbl
[163] V. A. Sadovnichii, “Dzeta-funktsiya i sobstvennye chisla differentsialnykh operatorov”, Differents. uravneniya, 10:7 (1974), 1276–1285 | MR | Zbl
[164] G. Illies, “Regularized products and determinants”, Comm. Math. Phys., 220:1 (2001), 69–94 | DOI | MR | Zbl
[165] B. V. Fedosov, “Analiticheskie formuly indeksa ellipticheskikh operatorov”, Tr. MMO, 30, 1974, 159–241 | MR | Zbl
[166] B. V. Fedosov, “Ob indekse ellipticheskikh semeistv na mnogoobrazii s kraem”, Dokl. AN SSSR, 248:5 (1979), 1066–1069 | MR | Zbl
[167] B. V. Fedosov, M. A. Shubin, “Indeks sluchainykh ellipticheskikh operatorov. II”, Matem. sb., 106(148):3 (1978), 455–483 | MR | Zbl
[168] Sh. Rempel, B.-V. Shultse, Teoriya indeksa ellipticheskikh kraevykh zadach, Mir, M., 1986 | MR | MR | Zbl
[169] D. Bollé, F. Gesztesy, H. Grosse, W. Schweiger, B. Simon, “Witten index, axial anomaly, and Krein's spectral shift function in supersymmetric quantum mechanics”, J. Math. Phys., 28:7 (1987), 1512–1525 | DOI | MR | Zbl
[170] F. Gesztesy, B. Simon, “Topological invariance of the Witten index”, J. Funct. Anal., 79:1 (1988), 91–102 | DOI | MR | Zbl
[171] L. A. Dikii, “Novyi sposob priblizhennogo vychisleniya sobstvennykh chisel zadachi Shturma–Liuvillya”, Dokl. AN SSSR, 116:1 (1957), 12–14 | MR | Zbl
[172] A. A. Dorodnitsyn, “Asimptoticheskie zakony raspredeleniya sobstvennykh znachenii dlya nekotorykh osobykh vidov differentsialnykh uravnenii vtorogo poryadka”, UMN, 7:6 (1952), 3–96 | MR | Zbl
[173] S. A. Shkarin, “O sposobe Gelfanda–Dikogo vychisleniya pervykh sobstvennykh znachenii operatora Shturma–Liuvillya”, Vestn. MGU. Ser. 1. Matem., mekh., 1996, no. 1, 39–44 | MR | Zbl
[174] V. A. Sadovnichii, V. E. Podolskii, “O needinstvennosti resheniya sistemy regulyarizovannykh sledov”, Dokl. RAN, 402:4 (2005), 455–456 | MR
[175] V. A. Sadovnichii, V. E. Podolskii, “Ob odnom klasse operatorov Shturma–Liuvillya i priblizhennom vychislenii pervykh sobstvennykh znachenii”, Matem. sb., 189:1 (1998), 133–148 | MR | Zbl
[176] V. A. Sadovnichii, V. V. Dubrovskii, E. M. Maleko, “Ob odnom sposobe priblizhennogo nakhozhdeniya sobstvennykh chisel operatora Shturma–Liuvillya”, Dokl. RAN, 369:1 (1999), 16–18 | MR | Zbl
[177] V. A. Sadovnichii, V. V. Dubrovskii, “Zamechaniya ob odnom novom metode vychisleniya sobstvennykh znachenii i sobstvennykh funktsii diskretnogo operatora”, Tr. sem. im. I. G. Petrovskogo, 17, 1994, 244–248 | MR | Zbl
[178] V. A. Sadovnichii, V. V. Dubrovskii, S. I. Kadchenko, V. F. Kravchenko, “Vychislenie pervykh sobstvennykh chisel kraevoi zadachi gidrodinamicheskoi ustoichivosti techeniya Puazeilya v krugloi trube”, Differents. uravneniya, 34:1 (1998), 50–53 | MR | Zbl
[179] V. A. Sadovnichii, V. V. Dubrovskii, S. I. Kadchenko, V. F. Kravchenko, “Vychislenie pervykh sobstvennykh chisel zadachi gidrodinamicheskoi ustoichivosti techeniya vyazkoi zhidkosti mezhdu dvumya vraschayuschimisya tsilindrami”, Differents. uravneniya, 36:6 (2000), 742–746 | MR | Zbl
[180] P. Nevai, “Orthogonal polynomials, recurrences, Jacobi matrices, and measures”, Progress in Approximation Theory (Tampa, FL, 1990), Springer Ser. Comput. Math., 19, Springer, New York, 1992, 79–104 | MR | Zbl
[181] B. P. Osilenker, “Analog formuly sleda dlya diskretnykh operatorov Shturma–Liuvillya s asimptoticheski $N$-periodicheskimi koeffitsientami”, Funkts. analiz i ego prilozh., 31:4 (1997), 72–75 | MR | Zbl
[182] R. Killip, B. Simon, “Sum rules for Jacobi matrices and their applications to spectral theory”, Ann. of Math. (2), 158:1 (2003), 253–321 | DOI | MR | Zbl
[183] R. Sh. Bakuradze, I. V. Stankevich, “O nekotorykh tozhdestvakh dlya sobstvennykh chisel i sobstvennykh funktsii odnogo klassa differentsialnykh operatorov chetvertogo poryadka”, Differents. uravneniya, 26:9 (1990), 1483–1490 | MR | Zbl