×
Badalov, V. H.; Badalov, S. V.

Generalised tanh-shaped hyperbolic potential: Klein-Gordon equation’s bound state solution. (English) Zbl 1519.81175

Commun. Theor. Phys. 75, No. 7, Article ID 075003, 11 p. (2023).

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Keywords:

Klein-Gordon equation; hyperbolic potential; Nikiforov-Uvarov method; diatomic molecules
Cite Review PDF
Full Text: DOI

References:

[1] Bagrov, V. G.; Gitman, D. M., Exact Solutions of Relativistic Wave Equations (1990), Dordrecht: Kluwer, Dordrecht · Zbl 0723.35066
[2] Greiner, W., Relativistic Quantum Mechanics. Wave Equations (1987), Berlin: Springer, Berlin · Zbl 0884.00015 · doi:10.1007/978-3-662-04275-5
[3] Dong, S-H, Factorization Method in Quantum Mechanics (2007), The Netherlands: Springer, The Netherlands · Zbl 1130.81001 · doi:10.1007/978-1-4020-5796-0
[4] Boivin, L.; Kärtner, F. X.; Haus, H. A., Analytical solution to the quantum field theory of self-phase modulation with a finite response time, Phys. Rev. Lett., 73, 240-243 (1994) · doi:10.1103/PhysRevLett.73.240
[5] Bialynicki-Birula, I., Particle beams guided by electromagnetic vortices: new solutions of the Lorentz, Schrödinger, Klein-Gordon, and Dirac equations, Phys. Rev. Lett., 93 (2004) · doi:10.1103/PhysRevLett.93.020402
[6] Belić, M.; Petrović, N.; Zhong, W. P.; Xie, R. H.; Chen, G., Analytical light bullet solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation, Phys. Rev. Lett., 101 (2008) · doi:10.1103/PhysRevLett.101.123904
[7] Garavelli, S. L.; Oliveira, F. A., Analytical solution for a Yukawa-type potential, Phys. Rev. Lett., 66, 1310-1313 (1991) · doi:10.1103/PhysRevLett.66.1310
[8] Flügge, S., Practical Quantum Mechanics (1999), Berlin: Springer, Berlin · Zbl 0934.81001 · doi:10.1007/978-3-642-61995-3
[9] Schneider, B.; Gharibnejad, H., Numerical methods every atomic and molecular theorist should know, Nat. Rev. Phys., 2, 89-102 (2020) · doi:10.1038/s42254-019-0126-3
[10] Witten, E., Dynamical breaking of supersymmetry, Nucl. Phys. B, 188, 513-554 (1981) · Zbl 1258.81046 · doi:10.1016/0550-3213(81)90006-7
[11] Kreyszig, E., Advanced Engineering Mathematics (1993), New York: Wiley, New York · Zbl 0803.00001
[12] Grosche, C., Conditionally solvable path integral problems, J. Phys. A: Math. Gen., 28, 5889-5902 (1995) · Zbl 0872.58016 · doi:10.1088/0305-4470/28/20/018
[13] Nikiforov, A. F.; Uvarov, V. B., Special Functions of Mathematical Physics (1988), Basel: Birkhäuser, Basel · Zbl 0694.33005 · doi:10.1007/978-1-4757-1595-8
[14] Ciftci, H.; Hall, R. L.; Saad, N., Asymptotic iteration method for eigenvalue problems, J. Phys. A: Math. Gen., 36, 11807-11816 (2003) · Zbl 1070.34113 · doi:10.1088/0305-4470/36/47/008
[15] Ciftci, H.; Hall, R. L.; Saad, N., Construction of exact solutions to eigenvalue problems by the asymptotic iteration method, J. Phys. A: Math. Gen., 38, 1147-1155 (2005) · Zbl 1069.34127 · doi:10.1088/0305-4470/38/5/015
[16] Ma, Z. Q.; Xu, B. W., Quantum correction in exact quantization rules, EPL, 69, 685-691 (2005) · doi:10.1209/epl/i2004-10418-8
[17] Qiang, W-C; Dong, S-H, Proper quantization rule, EPL, 89, 10003 (2010) · doi:10.1209/0295-5075/89/10003
[18] Serrano, F. A.; Cruz-Irisson, M.; Dong, S-H, Proper quantization rule as a good candidate to semiclassical quantization rules, Ann. Phys., 523, 771-782 (2011) · Zbl 1229.81094 · doi:10.1002/andp.201000144
[19] Polyakov, A., Quantum geometry of bosonic strings, Phys. Lett. B, 103, 207-210 (1981) · doi:10.1016/0370-2693(81)90743-7
[20] Vilenkin, A., Approaches to quantum cosmology, Phys. Rev. D, 50, 2581-2594 (1994) · doi:10.1103/PhysRevD.50.2581
[21] Socorro, J.; D’Oleire, M., Inflation from supersymmetric quantum cosmology, Phys. Rev. D, 82 (2010) · doi:10.1103/PhysRevD.82.044008
[22] Rebesh, A. P.; Lev, B. I., Analytical solutions of the classical and quantum cosmological models with an exponential potential, Phys. Rev. D, 100 (2019) · doi:10.1103/PhysRevD.100.123533
[23] Song, X. Q.; Wang, C. W.; Jia, C. S., Thermodynamic properties for the sodium dimer, Chem. Phys. Lett., 673, 50-55 (2017) · doi:10.1016/j.cplett.2017.02.010
[24] Cole, J. T.; Makris, K. G.; Musslimani, Z. H.; Christodoulides, D. N.; Rotter, S., Twofold \(####\) symmetry in doubly exponential optical lattices, Phys. Rev., 93 (2016) · Zbl 1376.35094 · doi:10.1103/PhysRevA.93.013803
[25] Pivano, A.; Dolocan, V. O., Analytical description of the topological interaction between magnetic domain walls in nanowires, Phys. Rev., 101 (2020) · doi:10.1103/PhysRevB.101.014438
[26] Dechant, A.; Kindermann, F.; Widera, A.; Lutz, E., Continuous-time random walk for a particle in a periodic potential, Phys. Rev. Lett., 123 (2019) · doi:10.1103/PhysRevLett.123.070602
[27] Hoff, D. E M., Mirror-symmetry violation in bound nuclear ground states, Nature, 580, 52-55 (2020) · doi:10.1038/s41586-020-2123-1
[28] Ruiz, R. F G., Spectroscopy of short-lived radioactive molecules, Nature, 581, 396-400 (2020) · doi:10.1038/s41586-020-2299-4
[29] Jia, C-S; Diao, Y-F; Liu, X-J; Wang, P-Q; Liu, J-Y; Zhang, G-D, Equivalence of the Wei potential model and Tietz potential model for diatomic molecules, J. Chem. Phys., 137 (2012) · doi:10.1063/1.4731340
[30] Morse, P. M., Diatomic molecules according to the wave mechanics. II. Vibrational levels, Phys. Rev., 34, 57-64 (1929) · JFM 55.0539.02 · doi:10.1103/PhysRev.34.57
[31] Berkdemir, C.; Han, J., Any l-state solutions of the Morse potential through the Pekeris approximation and Nikiforov-Uvarov method, Chem. Phys. Lett., 409, 203-207 (2005) · doi:10.1016/j.cplett.2005.05.021
[32] Hulthén, L., On the virtual state of the deuteron, Ark. Mat. Astron. Fys., 29B, 1 (1942) · Zbl 0028.13901
[33] Bayrak, O.; Kocak, G.; Boztosun, I., Any l-state solutions of the Hulthén potential by the asymptotic iteration method, J. Phys. A: Math. Gen., 39, 11521-11529 (2006) · Zbl 1100.81022 · doi:10.1088/0305-4470/39/37/012
[34] Domínguez-Adame, F., Bound states of the Klein-Gordon equation with vector and scalar Hulthén-type potentials, Phys. Lett. A, 136, 175-177 (1989) · doi:10.1016/0375-9601(89)90555-0
[35] Ahmadov, A. I.; Aslanova, S. M.; Sh, M.; Orujova, S. V.; Badalov; Dong, S-H, Approximate bound state solutions of the Klein-Gordon equation with the linear combination of Hulthén and Yukawa potentials, Phys. Lett. A, 383, 3010-3017 (2019) · Zbl 1476.81031 · doi:10.1016/j.physleta.2019.06.043
[36] Ahmadov, A. I.; Aslanova, S. M.; Orujova, M. S.; Badalov, S. V., Analytical bound state solutions of the Klein-Fock-Gordon equation for the sum of Hulthén and Yukawa potential within SUSY quantum mechanics, Adv. High Energy Phys., 2021 (2021) · Zbl 1457.81059 · doi:10.1155/2021/8830063
[37] Okon, I. B.; Popoola, O.; Isonguyo, C. N., Approximate solution of Schrodinger equation with some diatomic molecular interactions using Nikiforov-Uvarov method, Adv. High Energy Phys., 2017 (2017) · Zbl 1370.81062 · doi:10.1155/2017/9671816
[38] Okon, I. B.; Omugbe, E.; Antia, A. D.; Onate, C. A.; Akpabio, L. E.; Osafile, O. E., Spin and pseudospin solutions to Dirac equation and its thermodynamic properties using hyperbolic Hulthen plus hyperbolic exponential inversely quadratic potential, Sci. Rep., 11, 892 (2021) · doi:10.1038/s41598-020-77756-x
[39] Woods, R. D.; Saxon, D. S., Diffuse surface optical model for nucleon-nuclei scattering, Phys. Rev., 95, 577-578 (1954) · doi:10.1103/PhysRev.95.577
[40] Badalov, V. H.; Ahmadov, H. I.; Ahmadov, A. I., Analytical solutions of the Schrödinger equation with the Woods-Saxon potential for arbitrary l state, Int. J. Mod. Phys., 18, 631-641 (2009) · doi:10.1142/S0218301309012756
[41] Badalov, V. H.; Ahmadov, H. I.; Badalov, S. V., Any l-state analytical solutions of the Klein-Gordon equation for the Woods-Saxon potential, Int. J. Mod. Phys., 19, 1463-1475 (2010) · doi:10.1142/S0218301310015862
[42] Badalov, V. H.; Baris, B.; Uzun, K., Bound states of the D-dimensional Schrödinger equation for the generalized Woods-Saxon potential, Mod. Phys. Lett. A, 34 (2019) · Zbl 1412.81134 · doi:10.1142/S0217732319501074
[43] Otsuka, T.; Gade, A.; Sorlin, O.; Suzuki, T.; Utsuno, Y., Evolution of shell structure in exotic nuclei, Rev. Mod. Phys., 92 (2020) · doi:10.1103/RevModPhys.92.015002
[44] Rosen, N.; Morse, P. M., On the vibrations of polyatomic molecules, Phys. Rev., 42, 210-217 (1932) · JFM 58.1361.05 · doi:10.1103/PhysRev.42.210
[45] Taşkın, F., Approximate solutions of the Schrödinger equation for the Rosen-Morse potential including centrifugal term, Int. J. Theor. Phys., 48, 2692-2697 (2009) · Zbl 1175.81108 · doi:10.1007/s10773-009-0059-1
[46] Yi, L. Z.; Diao, Y. F.; Liu, J. Y.; Jia, C. S., Bound states of the Klein-Gordon equation with vector and scalar Rosen-Morse-type potentials, Phys. Lett. A, 333, 212-217 (2004) · Zbl 1123.81350 · doi:10.1016/j.physleta.2004.10.054
[47] Soylu, A.; Bayrak, O.; Boztosun, I., Exact solutions of Klein-Gordon equation with scalar and vector Rosen-Morse-type potentials, Chin. Phys. Lett., 25, 2754-2757 (2008) · doi:10.1088/0256-307X/25/8/006
[48] Gu, X-Y; Dong, S-H; Ma, Z-Q, Energy spectra for modified Rosen-Morse potential solved by the exact quantization rule, J. Phys. A: Math. Theor., 42, 035303 (2009) · Zbl 1154.81335 · doi:10.1088/1751-8113/42/3/035303
[49] Eckart, C., The penetration of a potential barrier by electrons, Phys. Rev., 35, 1303-1309 (1930) · JFM 56.0750.03 · doi:10.1103/PhysRev.35.1303
[50] Dong, S-H; Qiang, W-C; Sun, G-H; Bezerra, V. B., Analytical approximations to the l-wave solutions of the Schrödinger equation with the Eckart potential, J. Phys. A: Math. Theor., 40, 10535-10540 (2007) · Zbl 1120.81024 · doi:10.1088/1751-8113/40/34/010
[51] Liu, X-Y; Wei, G-F; Long, C-Y, Arbitrary wave relativistic bound state solutions for the Eckart potential, Int. J. Theor. Phys., 48, 463-470 (2009) · Zbl 1162.81366 · doi:10.1007/s10773-008-9821-z
[52] Manning, M. F.; Rosen, N., A potential function for the vibrations of diatomic molecules, Phys. Rev., 44, 951 (1933) · doi:10.1103/PhysRev.44.951
[53] Ahmadov, A. I.; Aydin, C.; Uzun, O., Analytical solutions of the Klein-Fock-Gordon equation with the Manning-Rosen potential plus a ring-shaped-like potential, Int. J. Mod. Phys., 29 (2014) · Zbl 1284.81100 · doi:10.1142/S0217751X1450002X
[54] Ahmadov, A. I.; Demirci, M.; Aslanova, S. M.; Mustamin, M. F., Arbitrary l-state solutions of the Klein-Gordon equation with the Manning-Rosen plus a class of Yukawa potentials, Phys. Lett. A, 384 (2020) · Zbl 1448.81300 · doi:10.1016/j.physleta.2020.126372
[55] Hamzavi, M.; Ikhdair, S. M.; Thylwe, K. E., Equivalence of the empirical shifted Deng-Fan oscillator potential for diatomic molecules, J. Math. Chem., 51, 227-238 (2013) · Zbl 1269.81224 · doi:10.1007/s10910-012-0075-x
[56] Chen, T.; Lin, S-R; Jia, C-S, Solutions of the Klein-Gordon equation with the improved Rosen-Morse potential energy model, Eur. Phys. J. Plus, 128, 69 (2013) · doi:10.1140/epjp/i2013-13069-1
[57] Sun, G-H; Aoki, M. A.; Dong, S-H, Quantum information entropies of the eigenstates for the Pöschl-Teller-like potential, Chin. Phys. B, 22 (2013) · doi:10.1088/1674-1056/22/5/050302
[58] Sun, G-H; Chen, C-Y; Taud, H.; Yáñez-Márquez, C.; Dong, S-H, Exact solutions of the 1D Schrödinger equation with the Mathieu potential, Phys. Lett. A, 384 (2020) · Zbl 1448.81313 · doi:10.1016/j.physleta.2020.126480
[59] Dong, Q.; Sun, G-H; Jing, J.; Dong New, S-H, Findings for two new type sine hyperbolic potentials, Phys. Lett. A, 383, 270-275 (2019) · Zbl 1428.82004 · doi:10.1016/j.physleta.2018.10.034
[60] Schiöberg, D., The energy eigenvalues of hyperbolical potential functions, Mol. Phys., 59, 1123-1137 (1986) · doi:10.1080/00268978600102631
[61] Ikhdair, S. M.; Sever, R., Improved analytical approximation to arbitrary l-state solutions of the Schrödinger equation for the hyperbolical potential, Ann Phys., 521, 189-197 (2009) · Zbl 1163.81303 · doi:10.1002/andp.20095210403
[62] Hu, X-T; Liu, J-Y; Jia, C-S, The \(####\) state of Cs_2 molecule, Comput. Theor. Chem., 1019, 137-140 (2013) · doi:10.1016/j.comptc.2013.06.020
[63] Wang, P-Q; Liu, J-Y; Zhang, L-H; Cao, S-Y; Jia, C-S, Improved expressions for the Schiöberg potential energy models for diatomic molecules, J. Mol. Spectrosc., 278, 23-26 (2012) · doi:10.1016/j.jms.2012.07.001
[64] Jia, C. S.; Zhang, L. H.; Wang, C. W., Thermodynamic properties for the lithium dimer, Chem. Phys. Lett., 667, 211 (2017) · doi:10.1016/j.cplett.2016.11.059
[65] Hua, W., Four-parameter exactly solvable potential for diatomic molecules, Phys. Rev., 42, 2524-2529 (1990) · doi:10.1103/PhysRevA.42.2524
[66] Tietz, T., Potential-energy function for diatomic molecules, J. Chem. Phys., 38, 3036-3037 (1963) · doi:10.1063/1.1733648
[67] Jia, C-S; Zhang, Y.; Zeng, X-L; Liang, L-T, Identity for the exponential-type molecule potentials and the supersymmetry shape invariance, Commun. Theor. Phys., 36, 641-646 (2001) · doi:10.1088/0253-6102/36/6/641
[68] Ikot, A. N.; Lütfüog˜lu, B. C.; Ngwueke, M. I.; Udoh, M. E.; Zare, S.; Hassanabadi, H., Klein-Gordon equation particles in exponential-type molecule potentials and their thermodynamic properties in D dimensions, Eur. Phys. J. Plus, 131, 419 (2016) · doi:10.1140/epjp/i2016-16419-5
[69] Fu, K-X; Wang, M.; Jia, C-S, Improved five-parameter exponential-type potential energy model for diatomic molecules, Commun. Theor. Phys., 71, 103-106 (2019) · doi:10.1088/0253-6102/71/1/103
[70] Tan, M-S; He, S.; Jia, C-S, Molecular spinless energies of the improved Rosen-Morse potential energy model in D dimensions, Eur. Phys. J. Plus, 129, 264 (2014) · doi:10.1140/epjp/i2014-14264-2
[71] Liu, J-Y; Hu, X-T; Jia, C-S, Molecular energies of the improved Rosen-Morse potential energy model, Can. J. Chem., 92, 40-44 (2014) · doi:10.1139/cjc-2013-0396
[72] Hu, X-T; Zhang, L-H; Jia, C-S, D-dimensional energies for cesium and sodium dimers, Can. J. Chem., 92, 386-391 (2014) · doi:10.1139/cjc-2013-0584
[73] Araújo, J. P.; Ballester, M. Y., A comparative review of 50 analytical representation of potential energy interaction for diatomic systems: 100 years of history, Int. J. Quantum Chem., 121 (2021) · doi:10.1002/qua.26808
[74] Wang, C-W; Peng, X-L; Liu, J-Y; Jiang, R.; Li, X-P; Liu, Y-S; Liu, S-Y; Wei, L-S; Zhang, L-H; Jia, C-S, A novel formulation representation of the equilibrium constant for water gas shift reaction, Int. J. Hydrogen Energy, 47, 27821-27838 (2022) · doi:10.1016/j.ijhydene.2022.06.105
[75] Okon, I. B.; Popoola, O. O.; Omugbe, E.; Antia, A. D.; Isonguyo, C. N.; Ituen, E. E., Thermodynamic properties and bound state solutions of Schrodinger equation with Mobius square plus screened-Kratzer potential using Nikiforov-Uvarov method, Comput. Theor. Chem., 1196 (2021) · doi:10.1016/j.comptc.2020.113132
[76] Okon, I. B.; Isonguyo, C. N.; Antia, A. D.; Ikot, A. N.; Popoola, O. O., Fisher and Shannon information entropies for a noncentral inversely quadratic plus exponential Mie-type potential, Commun. Theor. Phys., 72 (2020) · Zbl 1451.82021 · doi:10.1088/1572-9494/ab7ec9
[77] Jia, C-S; Wang, C-W; Zhang, L-H; Peng, X-L; Tang, H-M; Zeng, R., Enthalpy of gaseous phosphorus dimer, Chem. Eng. Sci., 183, 26-29 (2018) · doi:10.1016/j.ces.2018.03.009
[78] Jia, C-S; Zeng, R.; Peng, X-L; Zhang, L-H; Zhao, Y-L, Entropy of gaseous phosphorus dimer, Chem. Eng. Sci., 190, 1-4 (2018) · doi:10.1016/j.ces.2018.06.009
[79] Peng, X-L; Jiang, R.; Jia, C-S; Zhang, L-H; Zhao, Y-L, Gibbs free energy of gaseous phosphorus dimer, Chem. Eng. Sci., 190, 122-125 (2018) · doi:10.1016/j.ces.2018.06.027
[80] Ding, Q-C; Jia, C-S; Wang, C-W; Peng, X-L; Liu, J-Y; Zhang, L-H; Jiang, R.; Zhu, S-Y; Yuan, H.; Tang, H-X, Unified non-fitting formulation representation of thermodynamic properties for diatomic substances, J. Mol. Liq., 371 (2023) · doi:10.1016/j.molliq.2022.121088
[81] Liang, D-C; Zeng, R.; Wang, C-W; Ding, Q-C; Wei, L-S; Peng, X-L; Liu, J-Y; Yu, J.; Jia, C-S, Prediction of thermodynamic properties for sulfur dioxide, J. Mol. Liq., 352 (2022) · doi:10.1016/j.molliq.2022.118722
[82] Ding, Q-C; Jia, C-S; Liu, J-Z; Li, J.; Du, R-F; Liu, J-Y; Peng, X-L; Wang, C-W; Tang, H-X, Prediction of thermodynamic properties for sulfur dimer, Chem. Phys. Lett., 803 (2022) · doi:10.1016/j.cplett.2022.139844
[83] Ahmadov, H. I.; Dadashov, E. A.; Huseynova, N. S.; Badalov, V. H., Generalized tanh-shaped hyperbolic potential: bound state solution of Schrödinger equation, Eur. Phys. J. Plus, 136, 244 (2021) · doi:10.1140/epjp/s13360-021-01202-8
[84] Williams, B. W.; Poulios, D. P., A simple method for generating exactly solvable quantum mechanical potentials, Eur. J. Phys., 14, 222-226 (1993) · doi:10.1088/0143-0807/14/5/006
[85] Peña, J. J.; Morales, J.; García-Ravelo, J., Bound state solutions of Dirac equation with radial exponential-type potentials, J. Math. Phys., 58 (2017) · Zbl 1360.81149 · doi:10.1063/1.4979617
[86] Pekeris, C. L., The rotation-vibration coupling in diatomic molecules, Phys. Rev., 45, 98-103 (1934) · Zbl 0008.23001 · doi:10.1103/PhysRev.45.98
[87] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (1965), New York: Dover, New York
[88] Zhang, Y. P.; Cheng, C. H.; Kim, J. T.; Stanojevic, J.; Eyler, E. E., Dissociation energies of molecular hydrogen and the hydrogen molecular ion, Phys. Rev. Lett., 92 (2004) · doi:10.1103/PhysRevLett.92.203003
[89] Hajigeorgiou, P. G., An extended Lennard-Jones potential energy function for diatomic molecules: application to ground electronicstates, J. Mol. Spectrosc., 263, 101-110 (2010) · doi:10.1016/j.jms.2010.07.003
[90] Peña-Gallego, A.; Abreu, P. E.; Varandas, A. J C., MRCI calculation, scaling of the external correlation, and modeling of potential energy curves for HCl and OCl, J. Phys. Chem. A, 104, 6241-6246 (2000) · doi:10.1021/jp994036t
[91] Komasa, J.; Piszczatowski, K.; Yach, G.; Przybytek, M.; Jeziorski, B.; Pachucki, K., Quantum electrodynamics effects in rovibrational spectra of molecular hydrogen, J. Chem. Theory Comput., 7, 3105-3115 (2011) · doi:10.1021/ct200438t
[92] Salumbides, E. J.; Dickenson, G. D.; Ivanov, T. I.; Ubachs, W., QED effects in molecules: test on rotational quantum states of H_2, Phys. Rev. Lett., 107 (2011) · doi:10.1103/PhysRevLett.107.043005
[93] Stanke, M.; Kedziera, D.; Bubin, S.; Molski, M.; Adamowicz, L., Orbit-orbit relativistic corrections to the pure vibrational non-Born-Oppenheimer energies of H_2, J. Chem. Phys., 128 (2008) · doi:10.1063/1.2834926
[94] Dickenson, G. D.; Niu, M. L.; Salumbides, E. J.; Komasa, J.; Eikema, K. S E.; Pachucki, K.; Ubachs, W., Fundamental vibration of molecular hydrogen, Phys. Rev. Lett., 110 (2013) · doi:10.1103/PhysRevLett.110.193601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.