Vortex Structures inside Spherical Mesoscopic Superconductor Plus Magnetic Dipole

<div>The effective potential <svg height="14.7625pt" id="M181" style="vertical-align:-5.47417pt" version="1.1" viewbox="-0.0498162 -9.28833 36.8341 14.7625" width="36.8341pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M697 650H468L461 623L492 619C539 613 547 605 518 546C481 471 367 264 278 116H276C239 278 197 500 186 567C180 604 185 613 226 619L252 623L260 650H24L17 623C78 617 92 613 108 533L216 -11H247C365 200 515 462 560 529C616 612 624 615 689 623L697 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.332,3.132)"><path d="M357 391C357 416 325 451 270 451C241 451 179 431 146 400C109 365 97 334 97 307C97 248 143 213 196 180C250 146 261 123 261 101C261 75 237 40 190 40C153 40 110 72 86 109C81 116 68 119 55 112C37 102 24 87 24 66C24 30 85 -12 137 -12C229 -12 331 62 331 140C331 189 304 218 236 260C197 284 164 309 164 346C164 381 194 400 220 400C259 400 282 380 304 350C310 342 321 339 330 344C347 353 357 372 357 391Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.297,3.132)"><path d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,15.73,3.132)"><path d="M509 90L493 118C465 91 427 70 418 70S409 77 415 103L469 327C494 429 466 451 436 451C412 451 387 444 356 427C307 400 222 342 160 264H158L246 667C253 701 252 710 243 710C228 710 169 680 87 673L83 644H117C155 644 160 640 149 588L24 -4L32 -12C57 -4 86 4 108 9L143 187C198 275 324 386 375 386C394 386 399 367 385 311L331 87C315 22 322 -12 351 -12C387 -12 452 24 509 90Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.074,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,25.572,0)"><path d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"></path></g><g transform="matrix(.013,0,0,-0.013,32.153,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> in the radial part of the Schrödinger equation in spherical coordinates, Eq. (<a href="https://static.hindawi.com/articles/amp/volume-2018/8652151/figures/#EEq23">26</a>). Solid lines represent <svg height="8.55521pt" id="M182" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 26.5841 8.55521" width="26.5841pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M383 397C383 424 344 448 299 448C244 448 172 409 132 375C66 319 23 227 23 146C23 42 74 -12 146 -12C208 -12 298 30 359 103L343 124C315 95 248 48 192 48C145 48 111 85 111 163C111 228 129 294 151 330C171 363 201 401 241 401C275 401 302 384 325 356C332 347 339 344 348 348C373 360 383 381 383 397Z"></path></g><g transform="matrix(.013,0,0,-0.013,8.914,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,20.177,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g></svg> potentials, and dotted lines represent <svg height="8.32035pt" id="M183" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -8.11395 26.5841 8.32035" width="26.5841pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M383 397C383 424 344 448 299 448C244 448 172 409 132 375C66 319 23 227 23 146C23 42 74 -12 146 -12C208 -12 298 30 359 103L343 124C315 95 248 48 192 48C145 48 111 85 111 163C111 228 129 294 151 330C171 363 201 401 241 401C275 401 302 384 325 356C332 347 339 344 348 348C373 360 383 381 383 397Z"></path></g><g transform="matrix(.013,0,0,-0.013,8.914,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,20.177,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g></svg> potentials, for the same <span class="nowrap"><svg height="12.2741pt" id="M184" style="vertical-align:-3.638201pt" version="1.1" viewbox="-0.0498162 -8.6359 20.3251 12.2741" width="20.3251pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"></path></g><g transform="matrix(.013,0,0,-0.013,7.446,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.589,0)"><path d="M565 434L556 448L482 423L322 189L287 373C280 410 261 448 236 448C207 448 161 415 110 338L131 317C160 355 181 374 193 374C204 374 208 366 217 330C230 279 247 185 259 116L50 -175C18 -223 29 -260 41 -278H51C97 -198 207 -33 270 57L313 -165C328 -235 349 -261 376 -261S445 -244 510 -150L487 -127C444 -178 423 -186 411 -186C402 -186 398 -177 387 -130C371 -62 349 39 332 135C390 217 524 392 565 434Z"></path></g></svg>.</span></div>

Advances in Mathematical Physics

fig3

Figure 3

Figure 3: Vortex Structures inside Spherical Mesoscopic Superconductor Plus Magnetic Dipole 