• journal/journal.article
• aip/jmp
• /content/aip/journal/jmp/51/3/10.1063/1.3327835
1887
No data available.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
f

### New trace formulae for a quadratic pencil of the Schrödinger operator

Access full text Article
By Chuan Fu Yang1,a)
View Affiliations Hide Affiliations
Affiliations:
1 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094 Jiangsu, People’s Republic of China
a) Electronic mail: chuanfuyang@tom.com.
J. Math. Phys. 51, 033506 (2010)
/content/aip/journal/jmp/51/3/10.1063/1.3327835

### References

• By Chuan Fu Yang
• Source: J. Math. Phys. 51, 033506 ( 2010 );
1.
1.Ahlfors, L. , Complex Analysis (McGraw-Hill, New York, 1966).
2.
2.Cao, C. W. and Zhuang, D. W. , “Some trace formulas for the Schrödinger equation with energy-dependent potential,” Acta Math. Sci. 5, 233 (1985).
3.
3.Dikii, L. A. , “On the Gelfand-Levitan formula,” Usp. Mat. Nauk 8, 119 (1953) (in Russian).
4.
4.Dikii, L. A. , “The zeta function of an ordinary differential equation on a closed interval,” Izv. Akad. Nauk SSSR, Ser. Mat. 19, 187 (1955) (in Russian).
5.
5.Dubrovskii, V. V. , “Regularized trace of the Sturm-Liouville operator,” Differentsial’nye Uravneniya 16, 1127 (1980) (in Russian).
6.
6.Gasymov, M. G. and Guseinov, G. Sh. , “Determination of diffusion operator on spectral data,” SSSR Dokl. 37, 19 (1981).
7.
7.Gelfand, I. M. and Levitan, B. M. , “On a simple identity for eigenvalues of the differential operator of second order,” Dokl. Akad. Nauk SSSR 88, 593 (1953) (in Russian).
8.
8.Gesztesy, F. and Holden, H. , “On new trace formulae for Schrödinger operators,” Acta Appl. Math. 39, 315 (1995).
http://dx.doi.org/10.1007/BF00994640
9.
9.Gesztesy, F. and Holden, H. , in Multiparticle Quantum Scattering with Applications to Nuclear, Atomic and Molecular Physics, edited by D. G. Truhlar and B. Simon (Springer, New York, 1997), pp. 121145.
10.
10.Gesztesy, F. , Holden, H. , Simon, B. , and Zhao, Z. , “A trace formula for multidimensional Schrödinger operators,” J. Funct. Anal. 141, 449 (1996).
http://dx.doi.org/10.1006/jfan.1996.0137
11.
11.Gesztesy, F. , Holden, H. , Simon, B. , and Zhao, Z. , “Trace formulae and inverse spectral theory for Schrödinger operators,” Bull., New Ser., Am. Math. Soc. 29, 250 (1993).
http://dx.doi.org/10.1090/S0273-0979-1993-00431-2
12.
12.Guseinov, G. S. and Levitan, B. M. , “On trace formulas for Sturm-Liouville operators,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh. 1978, 40 (1978) (in Russian).
13.
13.Guseinov, I. M. and Nabiev, I. M. , “The inverse spectral problem for pencils of differential operators,” Mat. Sb. 198, 1579 (2007).
http://dx.doi.org/10.1070/SM2007v198n11ABEH003897
14.
14.Jaulent, M. and Jean, C. , “The inverse s-wave scattering problem for a class of potentials depending on energy,” Commun. Math. Phys. 28, 177 (1972).
http://dx.doi.org/10.1007/BF01645775
15.
15.Kaup, D. J. and Newell, A. C. , “An exact solution for a derivative nonlinear Schrödinger equation,” J. Math. Phys. 19, 798 (1978).
http://dx.doi.org/10.1063/1.523737
16.
16.Krein, M. G. , “On perturbation determinants and the trace formula for unitary and self-adjoint operators,” Dokl. Akad. Nauk SSSR 144, 268 (1962) (in Russian).
17.
17.Lax, P. D. , “Trace formulas for the Schrödinger operator,” Commun. Pure Appl. Math. 47, 503 (1994).
http://dx.doi.org/10.1002/cpa.3160470405
18.
18.Levitan, B. M. , “Calculation of a regularized trace for the Sturm-Liouville operator,” Russ. Math. Surveys 19, 161 (1964) (in Russian).
http://dx.doi.org/10.1070/RM1964v019n02ABEH001145
19.
19.Lidskii, V. B. and Sadovnichii, V. A. , “Regularized sums of the roots of a class of entire functions,” Dokl. Akad. Nauk SSSR 176, 259 (1967) (in Russian).
20.
20.Lyubishkin, V. A. and Podol’skii, V. E. , “On the integrability of regularized traces of differential operators,” Mat. Zametki 54, 33 (1993)
http://dx.doi.org/10.1007/BF01212842
20.Lyubishkin, V. A. and Podol’skii, V. E. , [Math. Notes 54, 790 (1993)].
http://dx.doi.org/10.1007/BF01212842
21.
21.Makin, A. S. , “Trace formulas for the Sturm-Liouville operator with regular boundary conditions,” Dokl. Math. 76, 702 (2007).
http://dx.doi.org/10.1134/S1064562407050171
22.
22.Marchenko, M. A. , Sturm-Liouville Operators and Their Applications (Naukova Dumka, Kiev, 1977) (in Russian).
23.
23.Nabiev, I. M. , “Multiplicities and relative position of eigenvalues of a quadratic pencil of Sturm-Liouville operators,” Math Notes 67, 309 (2000).
http://dx.doi.org/10.1007/BF02676667
24.
24.Papanicolaou, V. G. , “Trace formulas and the behaviour of large eigenvalues,” SIAM J. Math. Anal. 26, 218 (1995).
http://dx.doi.org/10.1137/S0036141092224601
25.
25.Sadovnichii, V. A. , “On the trace of the difference of two ordinary differential operators of higher order,” Differentsial’nye Uravneniya 2, 1611 (1966) (in Russian).
26.
26.Sadovnichii, V. A. and Dubrovskii, V. V. , “On relations for eigenvalues of discrete operators. Trace formulas of partial differential operators,” Differentsial’nye Uravneniya 13, 2034 (1977) (in Russian).
27.
27.Sadovnichii, V. A. and Lyubishkin, V. A. , “Trace formulas and perturbation theory,” Dokl. Akad. Nauk SSSR 300, 1064 (1988) (in Russian).
28.
28.Sadovnichii, V. A. and Podol’skii, V. E. , “Traces of differential operators,” Diff. Eq. 45, 477 (2009).
http://dx.doi.org/10.1134/S0012266109040028
29.
29.Savchuk, A. M. and Shkalikov, A. A. , “Sturm-Liouville operators with singular potentials,” Mat. Zametki 66, 897 (1999)
http://dx.doi.org/10.1007/BF02674332
29.Savchuk, A. M. and Shkalikov, A. A. , [Math. Notes 66, 741 (1999)].
http://dx.doi.org/10.1007/BF02674332
30.
30.Savchuk, A. M. and Shkalikov, A. A. , “Trace formula for Sturm-Liouville operators with singular potentials,” Math. Notes 69, 387 (2001).
http://dx.doi.org/10.1023/A:1010239626324
31.
31.Shevchenko, R. F. , “Regularization of the trace of the ordinary differential operator,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh. 1965, 28 (1965) (in Russian).
32.
32.Trubowitz, E. , “The inverse problem for periodic potentials,” Commun. Pure Appl. Math. 30, 321 (1977).
http://dx.doi.org/10.1002/cpa.3160300305
33.
33.Vinokurov, V. A. and Sadovnichii, V. A. , “The eigenvalue and the trace of the Sturm-Liouville operator as differentiable functions of an integrable potential,” Dokl. Ross. Akad. Nauk 365, 295 (1999) (in Russian).
34.
34.Yang, C. F. , “Reconstruction of the diffusion operator from nodal data,” Z. Naturforsch., A: Phys. Sci. 65a, 1 (2010).
http://aip.metastore.ingenta.com/content/aip/journal/jmp/51/3/10.1063/1.3327835
View: Figures

## Figures

Click to view

FIG. 1.

Contour in -complex plane.

/content/aip/journal/jmp/51/3/10.1063/1.3327835
2010-03-10
2013-12-08

/deliver/fulltext/aip/journal/jmp/51/3/1.3327835.html;jsessionid=1ur4gvsvrtfvu.x-aip-live-02?itemId=/content/aip/journal/jmp/51/3/10.1063/1.3327835&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/jmp

### Most read this month

Article
content/aip/journal/jmp
Journal
5
3

### Most cited this month

More Less
true
true
This is a required field